Chapter IV

Matrices

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Section 17Operations on matrices[29/79%]
Problem 17.1

Multiply the matrices:

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(a)

(1n01)⋅(1m01)\left(\begin{array}{cc}1 & n \\ 0 & 1\end{array}\right) \cdot \left(\begin{array}{cc}1 & m \\ 0 & 1\end{array}\right);

(b)

(cos⁡α−sin⁡αsin⁡αcos⁡α)⋅(cos⁡β−sin⁡βsin⁡βcos⁡β)\left(\begin{array}{rr}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right) \cdot \left(\begin{array}{rr}\cos \beta & -\sin \beta \\ \sin \beta & \cos \beta \end{array}\right);

(c)

(3−452−313−5−1)⋅(3292180−3)\left(\begin{array}{rrr}3 & -4 & 5 \\ 2 & -3 & 1 \\ 3 & -5 & -1\end{array}\right) \cdot \left(\begin{array}{cc}3 & 29 \\ 2 & 18 \\ 0 & -3\end{array}\right);

(d)

(1532−31)⋅(2−35−14−23−11)\left(\begin{array}{rrr}1 & 5 & 3 \\ 2 & -3 & 1\end{array}\right) \cdot \left(\begin{array}{rrr}2 & -3 & 5 \\ -1 & 4 & -2 \\ 3 & -1 & 1\end{array}\right);

(e)

(121313121)⋅(131212131)\left(\begin{array}{lll}1 & 2 & 1 \\ 3 & 1 & 3 \\ 1 & 2 & 1\end{array}\right) \cdot \left(\begin{array}{lll}1 & 3 & 1 \\ 2 & 1 & 2 \\ 1 & 3 & 1\end{array}\right);

(f)

(1−13−11−32−26)⋅(15203−121−1)\left(\begin{array}{rrr}1 & -1 & 3 \\ -1 & 1 & -3 \\ 2 & -2 & 6\end{array}\right) \cdot \left(\begin{array}{rrr}1 & 5 & 2 \\ 0 & 3 & -1 \\ 2 & 1 & -1\end{array}\right);

(g)

(1200210000130031)⋅(11001100001−100−11)\left(\begin{array}{llll}1 & 2 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 3 & 1\end{array}\right) \cdot \left(\begin{array}{rrrr}1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & -1 & 1\end{array}\right);

(h)

(1100120000310011)⋅(11001300001200−31)\left(\begin{array}{llll}1 & 1 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 1 & 1\end{array}\right) \cdot \left(\begin{array}{rrrr}1 & 1 & 0 & 0 \\ 1 & 3 & 0 & 0 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & -3 & 1\end{array}\right).

Problem 17.2

Carry out the operations:

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(a)

(302001212300)⋅(1−222−11−11−222−1)+(−20−306−35−28)\left(\begin{array}{llll}3 & 0 & 2 & 0 \\ 0 & 1 & 2 & 1 \\ 2 & 3 & 0 & 0\end{array}\right) \cdot \left(\begin{array}{rrr}1 & -2 & 2 \\ 2 & -1 & 1 \\ -1 & 1 & -2 \\ 2 & 2 & -1\end{array}\right)+\left(\begin{array}{rrr}-2 & 0 & -3 \\ 0 & 6 & -3 \\ 5 & -2 & 8\end{array}\right);

(b)

(302013220010)⋅(12−12−2−1122112)+(0−46122−5−22−2641301)\left(\begin{array}{lll}3 & 0 & 2 \\ 0 & 1 & 3 \\ 2 & 2 & 0 \\ 0 & 1 & 0\end{array}\right) \cdot \left(\begin{array}{rrrr}1 & 2 & -1 & 2 \\ -2 & -1 & 1 & 2 \\ 2 & 1 & 1 & 2\end{array}\right)+\left(\begin{array}{rrrr}0 & -4 & 6 & 1 \\ 2 & 2 & -5 & -2 \\ 2 & -2 & 6 & 4 \\ 1 & 3 & 0 & 1\end{array}\right).

Problem 17.3

Compute:

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(a)

(12221−22−21)2\left(\begin{array}{rrr}1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1\end{array}\right)^{2};

(b)

(0100001000010000)2\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{array}\right)^{2};

(c)

(111111−1−11−11−11−1−11)2\left(\begin{array}{rrrr}1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1\end{array}\right)^{2};

(d)

(0100002000030000)2\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0\end{array}\right)^{2}.

Problem 17.4

Compute:

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(a)

(cos⁡αsin⁡α−sin⁡αcos⁡α)n\left(\begin{array}{rr}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right)^{n};

(b)

(λ10λ)n\left(\begin{array}{cc}\lambda & 1 \\ 0 & \lambda \end{array}\right)^{n};

(c)

(2153)⋅(1011)⋅(3−1−52)n\left(\begin{array}{ll}2 & 1 \\ 5 & 3\end{array}\right) \cdot \left(\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right) \cdot \left(\begin{array}{rr}3 & -1 \\ -5 & 2\end{array}\right)^{n}.

Problem 17.5

Compute the value of the polynomial f(x)f(x) at the matrix AA:

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(a)

f(x)=x3−2x2+1,A=(210020111)f(x)=x^{3}-2 x^{2}+1, \quad A=\left(\begin{array}{ccc}2 & 1 & 0 \\ 0 & 2 & 0 \\ 1 & 1 & 1\end{array}\right);

(b)

f(x)=x3−3x+2,A=(211121112)f(x)=x^{3}-3 x+2, \quad A=\left(\begin{array}{lll}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2\end{array}\right).

Problem 17.6

Prove that if the matrices AA and BB commute, then

(A+B)n=∑i=0n(ni)AiBn−i (A+B)^{n}=\sum _{i=0}^{n}\binom {n}{i} A^{i} B^{n-i}

Give an example of two matrices A,BA, B for which this formula fails.

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Problem 17.7

Compute the powers of the square matrix

H=[010…0001…0……………000…1000…0] H=\left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 \\ 0 & 0 & 0 & \ldots & 0 \end{smallmatrix}\right]
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Problem 17.8

Let a square matrix

J=[λ10…000λ1…00………………000…λ1000…0λ] J=\left[\begin{smallmatrix} \lambda & 1 & 0 & \ldots & 0 & 0 \\ 0 & \lambda & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & \lambda & 1 \\ 0 & 0 & 0 & \ldots & 0 & \lambda \end{smallmatrix}\right]

of size nn be given. Prove that if f(x)f(x) is a polynomial, then

f(J)=[f(λ)f′(λ)1!f′′(λ)2!…f(n−2)(λ)(n−2)!f(n−1)(λ)(n−1)!0f(λ)f′(λ)1!…f(n−3)(λ)(n−3)!f(n−2)(λ)(n−2)!………………………………………………………………f′(λ)1!000…f(λ)f′(λ)1!000…0f(λ)]. f(J)=\left[\begin{smallmatrix} f(\lambda ) & \frac{f^{\prime }(\lambda )}{1!} & \frac{f^{\prime \prime }(\lambda )}{2!} & \ldots & \frac{f^{(n-2)}(\lambda )}{(n-2)!} & \frac{f^{(n-1)}(\lambda )}{(n-1)!} \\ 0 & f(\lambda ) & \frac{f^{\prime }(\lambda )}{1!} & \ldots & \frac{f^{(n-3)}(\lambda )}{(n-3)!} & \frac{f^{(n-2)}(\lambda )}{(n-2)!} \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots & \ldots \ldots \ldots \ldots \ldots \ldots & \frac{f^{\prime }(\lambda )}{1!} \\ 0 & 0 & 0 & \ldots & f(\lambda ) & \frac{f^{\prime }(\lambda )}{1!} \\ 0 & 0 & 0 & \ldots & 0 & f(\lambda ) \end{smallmatrix}\right].
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Problem 17.9

Let C,AC, A be square matrices of the same size and let f(x)f(x) be a polynomial. Prove that f(CAC−1)=Cf(A)C−1f\left(C A C^{-1}\right)=C f(A) C^{-1}.

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Problem 17.10

Compute eAe^{A}, where:

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(a)

A=(21−4−2)A=\left(\begin{array}{rr}2 & 1 \\ -4 & -2\end{array}\right);

(b)

A=(012006000)A=\left(\begin{array}{lll}0 & 1 & 2 \\ 0 & 0 & 6 \\ 0 & 0 & 0\end{array}\right).

Problem 17.11

Compute ln⁡A\ln A, where:

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(a)

A=(31−4−1)A=\left(\begin{array}{rr}3 & 1 \\ -4 & -1\end{array}\right);

(b)

A=(110…0011…0……………000…1)A=\left(\begin{array}{ccccc}1 & 1 & 0 & \ldots & 0 \\ 0 & 1 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1\end{array}\right).

Problem 17.12

Let A=(aij)A=\left(a_{i j}\right) be a matrix of size m×nm \times n. Prove that

A=∑i,jaijEij A=\sum _{i, j} a_{i j} E_{i j}

where EijE_{i j} are the matrix units.

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Problem 17.13

Prove that EijEpq=δjpEiqE_{i j} E_{p q}=\delta_{j p} E_{i q}.

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Problem 17.14

Let AA be an arbitrary matrix. Compute EijAE_{i j} A.

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Problem 17.15

Let AA be an arbitrary matrix. Compute AEijA E_{i j}.

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Problem 17.16

Let AA be a square matrix such that EijA=AEijE_{i j} A=A E_{i j} for all matrix units EijE_{i j}. Prove that A=λEA=\lambda E for some scalar λ\lambda.

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Problem 17.17

Let AA be a square matrix such that EiiA=AEiiE_{i i} A=A E_{i i} for all ii. Prove that the matrix AA is diagonal.

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Problem 17.18

Let a square matrix AA commute with all nonsingular matrices. Prove that A=λEA=\lambda E.

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Problem 17.19

Find all matrices AA of order nn such that tr⁡AX=0\operatorname {tr} A X=0 for any matrix XX of order nn.

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Problem 17.20

Prove that the trace of the product of two matrices does not depend on the order of the factors.

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Problem 17.21

Prove that if CC is a nonsingular matrix, then for any matrix AA of the same order tr⁡CAC−1=tr⁡A\operatorname {tr} C A C^{-1}=\operatorname {tr} A.

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Problem 17.22

For which λ\lambda does the equation

[X,Y]=λE [X, Y]=\lambda E

have a solution ([X,Y][X, Y] is the commutator of the matrices XX and YY)?

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Problem 17.23

Prove that for any square matrices A,B,CA, B, C the following equalities hold:

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(a)

[A,BC]=[A,B]C+B[A,C][A, B C]=[A, B] C+B[A, C];

(b)

[[A,B],C]+[[B,C],A]+[[C,A],B]=0[[A, B], C]+[[B, C], A]+[[C, A], B]=0.

Problem 17.24

Prove that for any matrices of order 2 the equality [[A,B]2,C]=0\left[[A, B]^{2}, C\right]=0 holds.

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Problem 17.25

Let A,B,…,D1A, B, \ldots , D_{1} be square matrices of the same order. Express the product of the matrices

[ABCD]⋅[A1B1C1D1] \left[\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}\right] \cdot \left[\begin{smallmatrix} A_{1} & B_{1} \\ C_{1} & D_{1} \end{smallmatrix}\right]

in terms of the given matrices.

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Problem 17.26

Let AA be a real triangular matrix commuting with tA{ }^{t} A. Prove that the matrix AA is diagonal.

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Problem 17.27

Let A=(aij)∈Mn(R)A=\left(a_{i j}\right) \in \mathbf{M}_{n}(\mathbb {R}) be a symmetric nonsingular matrix such that there exists k<nk<n for which aij=0a_{i j}=0 whenever ∣i−j∣⩾k\left|i-j\right| \geqslant k. Suppose that A=tB⋅BA={ }^{t} B \cdot B, where B=(bij)B=\left(b_{i j}\right) is an upper triangular matrix. Prove that bij=0b_{i j}=0 if j−i⩾kj-i \geqslant k.

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Problem 17.28

Prove that any matrix with trace 0 is a sum of commutators of matrices with trace 0.

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Problem 17.29

For the matrix

[010…0001…0……………000…1000…0] \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 \\ 0 & 0 & 0 & \ldots & 0 \end{smallmatrix}\right]

find matrices AA and BB such that

[A,X]=X,[A,B]=−B,[X,B]=A [A, X]=X, \quad [A, B]=-B, \quad [X, B]=A
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Section 18Matrix equations. Inverse matrix[24/83%]
Problem 18.1

Solve the system of matrix equations:

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(a)

X+Y=(1101),2X+3Y=(1001);X+Y=\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right), \quad 2 X+3 Y=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right) ;

(b)

2X−Y=(01−10),−4X+2Y=(0−220)2 X-Y=\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right), \quad -4 X+2 Y=\left(\begin{array}{rr}0 & -2 \\ 2 & 0\end{array}\right).

Problem 18.2

Prove that a square matrix XX of order 2 satisfies the equation

X2−(tr⁡X)X+det⁡X=0 X^{2}-(\operatorname {tr} X) X+\operatorname {det} X=0
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Problem 18.3

Solve the matrix equations

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(a)

(1312)X=(1111)\left(\begin{array}{ll}1 & 3 \\ 1 & 2\end{array}\right) X=\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right);

(b)

X(−113−4)=(−2−134)X\left(\begin{array}{rr}-1 & 1 \\ 3 & -4\end{array}\right)=\left(\begin{array}{rr}-2 & -1 \\ 3 & 4\end{array}\right);

(c)

(2−14−2)X=(1326);\left(\begin{array}{ll}2 & -1 \\ 4 & -2\end{array}\right) X=\left(\begin{array}{ll}1 & 3 \\ 2 & 6\end{array}\right) ; \quad

(d)

X(2−14−2)=(1362);X\left(\begin{array}{ll}2 & -1 \\ 4 & -2\end{array}\right)=\left(\begin{array}{ll}1 & 3 \\ 6 & 2\end{array}\right) ;

(e)

(3121)X(1312)=(3322)\left(\begin{array}{ll}3 & 1 \\ 2 & 1\end{array}\right) X\left(\begin{array}{ll}1 & 3 \\ 1 & 2\end{array}\right)=\left(\begin{array}{ll}3 & 3 \\ 2 & 2\end{array}\right);

(f)

(12−332−42−10)X=(1−3010271078)\left(\begin{array}{rrr}1 & 2 & -3 \\ 3 & 2 & -4 \\ 2 & -1 & 0\end{array}\right) X=\left(\begin{array}{rrr}1 & -3 & 0 \\ 10 & 2 & 7 \\ 10 & 7 & 8\end{array}\right);

(g)

X(5311−3−2−521)=(−830−590−2150)X\left(\begin{array}{rrr}5 & 3 & 1 \\ 1 & -3 & -2 \\ -5 & 2 & 1\end{array}\right)=\left(\begin{array}{ccc}-8 & 3 & 0 \\ -5 & 9 & 0 \\ -2 & 15 & 0\end{array}\right);

(h)

(11…101…1……….00…1)X=(123…n012…n−1……………000…1)\left(\begin{array}{cccc}1 & 1 & \ldots & 1 \\ 0 & 1 & \ldots & 1 \\ \ldots & \ldots & \ldots & . \\ 0 & 0 & \ldots & 1\end{array}\right) X=\left(\begin{array}{ccccc}1 & 2 & 3 & \ldots & n \\ 0 & 1 & 2 & \ldots & n-1 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1\end{array}\right);

(i)

X(11−12101−11)=(1−134321−25)X\left(\begin{array}{rrr}1 & 1 & -1 \\ 2 & 1 & 0 \\ 1 & -1 & 1\end{array}\right)=\left(\begin{array}{rrr}1 & -1 & 3 \\ 4 & 3 & 2 \\ 1 & -2 & 5\end{array}\right);

(j)

(121212123)X=(210112−121)\left(\begin{array}{lll}1 & 2 & 1 \\ 2 & 1 & 2 \\ 1 & 2 & 3\end{array}\right) X=\left(\begin{array}{rrr}2 & 1 & 0 \\ 1 & 1 & 2 \\ -1 & 2 & 1\end{array}\right);

(k)

(210120001)X(001010100)=(010100000)\left(\begin{array}{lll}2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 1\end{array}\right) X\left(\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right)=\left(\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{array}\right);

(l)

X(111123149)=(123246369)X\left(\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9\end{array}\right)=\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9\end{array}\right);

(m)

(123231312)X=(001100010)\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2\end{array}\right) X=\left(\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right);

(n)

(110212011)X=(5−12−646−207)\left(\begin{array}{lll}1 & 1 & 0 \\ 2 & 1 & 2 \\ 0 & 1 & 1\end{array}\right) X=\left(\begin{array}{rrr}5 & -1 & 2 \\ -6 & 4 & 6 \\ -2 & 0 & 7\end{array}\right).

Problem 18.4

Let A,BA, B be matrices of sizes m×nm \times n and m×km \times k respectively. Prove that the matrix equation AX=BA X=B, where XX is a matrix of size n×kn \times k, has a solution if and only if the rank of the matrix AA coincides with the rank of the augmented matrix (A∣B)(A \mid B).

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Problem 18.5

Let AA be a square matrix. Prove that the matrix equation AX=BA X=B has a unique solution if and only if the matrix AA is nonsingular.

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Problem 18.6

Let AA be a matrix of size n×mn \times m with m≠nm \neq n. Prove that for any natural number kk there exists a matrix BB of size n×kn \times k such that the matrix equation AX=BA X=B does not have a unique solution.

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Problem 18.7

Prove that the system of equations

∑j=1naijXj=Bi,i=1,2,…,n \sum _{j=1}^{n} a_{i j} X_{j}=B_{i}, \quad i=1,2, \ldots , n

where XjX_{j} and BiB_{i} are matrices of size p×qp \times q, has a unique solution if and only if det⁡(aij)≠0\operatorname {det}\left(a_{i j}\right) \neq 0.

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Problem 18.8

Using the adjugate matrix, find the inverse of the matrix:

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(a)

(1301)\left(\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right);

(b)

(1032)\left(\begin{array}{ll}1 & 0 \\ 3 & 2\end{array}\right);

(c)

(1235)\left(\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right);

(d)

(1327)\left(\begin{array}{ll}1 & 3 \\ 2 & 7\end{array}\right);

(e)

(50003000−2)\left(\begin{array}{rrr}5 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -2\end{array}\right);

(f)

(100010301)\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 3 & 0 & 1\end{array}\right);

(g)

(600012035)\left(\begin{array}{lll}6 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 3 & 5\end{array}\right);

(h)

(130270007)\left(\begin{array}{lll}1 & 3 & 0 \\ 2 & 7 & 0 \\ 0 & 0 & 7\end{array}\right);

(i)

(110010033)\left(\begin{array}{lll}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 3 & 3\end{array}\right);

(j)

(200311002)\left(\begin{array}{lll}2 & 0 & 0 \\ 3 & 1 & 1 \\ 0 & 0 & 2\end{array}\right);

(k)

(cos⁡α−sin⁡αsin⁡αcos⁡α)\left(\begin{array}{rr}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right).

Problem 18.9

Using elementary operations, find the inverse of the matrix:

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(a)

(1000001000010100);\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0\end{array}\right) ;

(b)

(0010100000010100)\left(\begin{array}{llll}0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0\end{array}\right);

(c)

(2000000102000010)\left(\begin{array}{llll}2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right);

(d)

(000−1002010000300)\left(\begin{array}{rrrr}0 & 0 & 0 & -1 \\ 0 & 0 & 2 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0\end{array}\right);

(e)

(11…101…1……….00…1)\left(\begin{array}{cccc}1 & 1 & \ldots & 1 \\ 0 & 1 & \ldots & 1 \\ \ldots & \ldots & \ldots & . \\ 0 & 0 & \ldots & 1\end{array}\right);

(f)

(100…00110…00011…00………………000…11)\left(\begin{array}{cccccc}1 & 0 & 0 & \ldots & 0 & 0 \\ 1 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 & 1\end{array}\right);

(g)

(2576345−2−3)\left(\begin{array}{rrr}2 & 5 & 7 \\ 6 & 3 & 4 \\ 5 & -2 & -3\end{array}\right);

(h)

(3−452−313−5−1)\left(\begin{array}{rrr}3 & -4 & 5 \\ 2 & -3 & 1 \\ 3 & -5 & -1\end{array}\right);

(i)

(273394153)\left(\begin{array}{lll}2 & 7 & 3 \\ 3 & 9 & 4 \\ 1 & 5 & 3\end{array}\right);

(j)

(12221−22−21);\left(\begin{array}{rrr}1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1\end{array}\right) ;

(k)

(12342312111−110−2−6)\left(\begin{array}{rrrr}1 & 2 & 3 & 4 \\ 2 & 3 & 1 & 2 \\ 1 & 1 & 1 & -1 \\ 1 & 0 & -2 & -6\end{array}\right).

Problem 18.10

Find the inverse of the square matrix:

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(a)

(A0BC)\left(\begin{array}{ll}A & 0 \\ B & C\end{array}\right);

(b)

(AB0C)\left(\begin{array}{cc}A & B \\ 0 & C\end{array}\right); where A,CA, C are nonsingular matrices.

Problem 18.11

Find the inverse of the matrix:

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(a)

(120023001−1130102)\left(\begin{array}{rrrr}1 & 2 & 0 & 0 \\ 2 & 3 & 0 & 0 \\ 1 & -1 & 1 & 3 \\ 0 & 1 & 0 & 2\end{array}\right);

(b)

(23121120001−1001−2)\left(\begin{array}{rrrr}2 & 3 & 1 & 2 \\ 1 & 1 & 2 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 1 & -2\end{array}\right).

Problem 18.12

Let A,B,C,DA, B, C, D be nonsingular matrices. Prove that

[ABCD]−1=[(A−BD−1C)−1(C−DB−1A)−1(B−AC−1D)−1(D−CA−1B)−1] \left[\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}\right]^{-1}=\left[\begin{smallmatrix} \left(A-B D^{-1} C\right)^{-1} & \left(C-D B^{-1} A\right)^{-1} \\ \left(B-A C^{-1} D\right)^{-1} & \left(D-C A^{-1} B\right)^{-1} \end{smallmatrix}\right]
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Problem 18.13

What values can the determinant take of

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(a)

an orthogonal matrix;

(b)

a unitary matrix?

Problem 18.14

What is the determinant of an integer matrix AA equal to, if the matrix A−1A^{-1} is also an integer matrix?

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Problem 18.15

Let AA be a square matrix of order nn whose entries are polynomials in the variable tt, and let det⁡A\operatorname {det} A be a nonzero polynomial. Prove that there exists a unique matrix BB, whose entries are polynomials in tt, such that AB=BA=(det⁡A)EA B=B A=(\operatorname {det} A) E. Find BB if:

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(a)

A=(1−t1+t1+t2t3)A=\left(\begin{array}{cc}1-t & 1+t \\ 1+t^{2} & t^{3}\end{array}\right);

(b)

A=(t1t−111−t1t)A=\left(\begin{array}{ccc}t & 1 & t \\ -1 & 1 & 1 \\ -t & 1 & t\end{array}\right).

Problem 18.16

Prove that in the ring of matrices over a field:

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(a)

an invertible matrix is not a zero divisor;

(b)

any matrix is either invertible or is both a left and a right zero divisor.

Problem 18.17

Prove that if the matrix E+ABE+A B is invertible, then the matrix E+BAE+B A is also invertible.

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Problem 18.18

Let AA and BB be matrices of sizes n×mn \times m and m×nm \times n respectively, such that ABA B and BAB A are identity matrices of sizes nn and mm. Prove that m=nm=n.

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Problem 18.19

Let AA be a matrix of size m×nm \times n having rank mm. Prove that there exists a matrix XX of size n×mn \times m such that AXA X is the identity matrix of size mm.

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Problem 18.20

How does the matrix A−1A^{-1} change if in AA:

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(a)

the ii-th and jj-th rows are swapped;

(b)

the jj-th row, multiplied by cc, is added to the ii-th row;

(c)

the ii-th row is multiplied by a number c≠0c \neq 0;

(d)

the transformations a)-c) are performed on columns?

Problem 18.21

Prove that (AB)−1=B−1A−1(A B)^{-1}=B^{-1} A^{-1}.

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Problem 18.22

Let X^\widehat{X} be the adjugate matrix (see 16.4) for a square matrix XX. Prove that

AB^=B^A^,A−1^=(A^)−1,(tA)^=t(A^) \widehat{A B}=\widehat{B} \widehat{A}, \quad \widehat{A^{-1}}=(\widehat{A})^{-1}, \quad \widehat{\left({ }^{t} A\right)}={ }^{t}(\widehat{A})
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Problem 18.23

Let BB and CC be rows of length nn such that CtB≠−1C^{t} B \neq -1, and let EE be the identity matrix of size nn. Prove that the matrix E+tBCE+{ }^{t} B C is invertible.

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Problem 18.24

Let B,CB, C be rows of length nn such that CtB=−1C^{t} B=-1, and let EE be the identity matrix of size nn. Prove that the rank of the matrix E+tBCE+{ }^{t} B C equals n−1n-1.

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Section 19Special matrices[28/54%]
Problem 19.1

Prove that Eii−Ejj=[Eij,Eji]E_{i i}-E_{j j}=\left[E_{i j}, E_{j i}\right] for i≠ji \neq j.

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Problem 19.2

Represent the matrix as a product of elementary matrices:

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(a)

(1245)\left(\begin{array}{ll}1 & 2 \\ 4 & 5\end{array}\right);

(b)

(011101110)\left(\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right).

Problem 19.3

Using the properties of elementary matrices, multiply the matrices:

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(a)

(1234135712481111)⋅(1000020000300004);\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 1 & 3 & 5 & 7 \\ 1 & 2 & 4 & 8 \\ 1 & 1 & 1 & 1\end{array}\right) \cdot \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4\end{array}\right) ;

(b)

(1000020000300004)⋅(1234135712481111)\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4\end{array}\right) \cdot \left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 1 & 3 & 5 & 7 \\ 1 & 2 & 4 & 8 \\ 1 & 1 & 1 & 1\end{array}\right);

(c)

(1234135712481111)⋅(100001002010−3001)\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 1 & 3 & 5 & 7 \\ 1 & 2 & 4 & 8 \\ 1 & 1 & 1 & 1\end{array}\right) \cdot \left(\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 1 & 0 \\ -3 & 0 & 0 & 1\end{array}\right);

(d)

(100011002010−3001)⋅(1234135712481111)\left(\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 2 & 0 & 1 & 0 \\ -3 & 0 & 0 & 1\end{array}\right) \cdot \left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 1 & 3 & 5 & 7 \\ 1 & 2 & 4 & 8 \\ 1 & 1 & 1 & 1\end{array}\right).

Problem 19.4

Prove the following properties of the transpose operation:

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(a)

t(A+B)=tA+tB{ }^{t}(A+B)={ }^{t} A+{ }^{t} B;

(b)

t(λA)=λtA{ }^{t}(\lambda A)=\lambda^{t} A;

(c)

t(AB)=tB⋅tA{ }^{t}(A B)={ }^{t} B \cdot { }^{t} A;

(d)

(tA)−1=t(A−1)\left({ }^{t} A\right)^{-1}={ }^{t}\left(A^{-1}\right);

(e)

t(tA)=A{ }^{t}\left({ }^{t} A\right)=A.

Problem 19.5

Prove that any matrix can be uniquely represented as the sum of a symmetric matrix and a skew-symmetric matrix.

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Problem 19.6

Prove that:

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(a)

if the matrices AA and BB are orthogonal, then the matrices A−1A^{-1} and ABA B are also orthogonal;

(b)

if the complex matrices AA and BB are unitary, then the matrices A−1A^{-1} and ABA B are also unitary.

Problem 19.7

Prove that:

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(a)

the product of two symmetric or two skew-symmetric matrices is a symmetric matrix if and only if these matrices commute;

(b)

the product of a symmetric and a skew-symmetric matrix is a skew-symmetric matrix if and only if these matrices commute.

Problem 19.8

Under what condition is the product of two Hermitian or two skew-Hermitian matrices a Hermitian matrix?

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Problem 19.9

Prove that for any square complex matrix XX there exists a matrix YY such that XYX=X,YXY=YX Y X=X, Y X Y=Y, and the matrices XYX Y and YXY X are Hermitian.

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Problem 19.10

Prove that the inverse of a symmetric or skew-symmetric matrix is symmetric or skew-symmetric, respectively.

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Problem 19.11

Prove that if the matrices AA and BB are both symmetric or both skew-symmetric, then their commutator [A,B][A, B] is a skew-symmetric matrix.

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Problem 19.12

Is it true that every skew-symmetric matrix is a sum of commutators of skew-symmetric matrices?

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Problem 19.13

Find all symmetric orthogonal and skew-symmetric orthogonal matrices of order 2.

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Problem 19.14

Find all lower nil-triangular matrices that commute with all lower nil-triangular matrices of the same order.

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Problem 19.15

Prove that the sum of two commuting nilpotent matrices is a nilpotent matrix. Is this statement true if the matrices do not commute?

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Problem 19.16

Prove that if the matrices A,BA, B and [A,B][A, B] are nilpotent and the matrices AA and BB commute with the matrix [A,B][A, B], then the matrix A+BA+B is nilpotent.

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Problem 19.17

Prove that if a matrix AA of order 2 is nilpotent, then A2=0A^{2}=0.

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Problem 19.18

Prove that every lower nil-triangular matrix is nilpotent.

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Problem 19.19

Prove that if a matrix AA is nilpotent, then the matrices E−AE-A and E+AE+A are invertible.

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Problem 19.20

Prove that if a matrix AA is nilpotent and the polynomial f(t)f(t) has a nonzero constant term, then the matrix f(A)f(A) is invertible.

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Problem 19.21

Solve the equation AX+X+A=0A X+X+A=0, where AA is a nilpotent matrix.

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Problem 19.22

Prove that a nilpotent matrix of order 2 has zero trace.

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Problem 19.23

Prove that the product of two commuting periodic matrices is a periodic matrix. Is this statement true if the matrices do not commute?

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Problem 19.24

Prove that the matrix CAC−1C A C^{-1} is nilpotent or periodic if and only if the matrix AA is nilpotent or periodic, respectively.

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Problem 19.25

Let σ\sigma be a permutation of the set {1,2,…,n}\left\{ 1,2, \ldots , n\right\} and let Aσ=(δiσ(j))(δijA_{\sigma }=\left(\delta_{i \sigma (j)}\right)\left(\delta_{i j}\right. being the Kronecker symbol). Prove that:

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(a)

the matrix AσA_{\sigma } is periodic;

(b)

for any permutations σ\sigma and τ\tau, Aστ=AσAτA_{\sigma \tau }=A_{\sigma } A_{\tau };

(c)

AσA_{\sigma } can be represented as a product of at most n−1n-1 elementary matrices.

Problem 19.26

Prove that the product of upper triangular matrices is an upper triangular matrix.

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Problem 19.27

Prove that the inverse of a unitriangular matrix is unitriangular.

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Problem 19.28

Let A,B,C,DA, B, C, D be square complex matrices of size nn, such that CD=DCC D=D C. Prove that

det⁡[ABCD]≠0⟺det⁡(AD−BC)≠0 \operatorname {det}\left[\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}\right] \neq 0 \Longleftrightarrow \operatorname {det}(A D-B C) \neq 0
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