Math Problems Vault

Problem 18.1

(a)(b)

Solve the system of matrix equations:

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(a)

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

(b)

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

Problem 18.2

Prove that a square matrix $X$ of order 2 satisfies the equation

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)

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

(b)

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

(c)

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

(d)

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

(e)

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

(f)

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

(g)

$X\left(\begin{array}{rrr}5 & 3 & 1 \\ 1 & -3 & -2 \\ -5 & 2 & 1\end{array}\right)=\left(\begin{array}{ccc}-8 & 3 & 0 \\ -5 & 9 & 0 \\ -2 & 15 & 0\end{array}\right)$ ;

(h)

$\left(\begin{array}{cccc}1 & 1 & \ldots & 1 \\ 0 & 1 & \ldots & 1 \\ \ldots & \ldots & \ldots & . \\ 0 & 0 & \ldots & 1\end{array}\right) X=\left(\begin{array}{ccccc}1 & 2 & 3 & \ldots & n \\ 0 & 1 & 2 & \ldots & n-1 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1\end{array}\right)$ ;

(i)

$X\left(\begin{array}{rrr}1 & 1 & -1 \\ 2 & 1 & 0 \\ 1 & -1 & 1\end{array}\right)=\left(\begin{array}{rrr}1 & -1 & 3 \\ 4 & 3 & 2 \\ 1 & -2 & 5\end{array}\right)$ ;

(j)

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

(k)

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

(l)

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

(m)

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

(n)

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

Problem 18.4

Let $A, B$ be matrices of sizes $m \times n$ and $m \times k$ respectively. Prove that the matrix equation $A X=B$ , where $X$ is a matrix of size $n \times k$ , has a solution if and only if the rank of the matrix $A$ coincides with the rank of the augmented matrix $(A \mid B)$ .

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Problem 18.5

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

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Problem 18.6

Let $A$ be a matrix of size $n \times m$ with $m \neq n$ . Prove that for any natural number $k$ there exists a matrix $B$ of size $n \times k$ such that the matrix equation $A X=B$ does not have a unique solution.

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Problem 18.7

Prove that the system of equations

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

where $X_{j}$ and $B_{i}$ are matrices of size $p \times q$ , has a unique solution if and only if $\operatorname {det}\left(a_{i j}\right) \neq 0$ .

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Problem 18.8

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

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(a)

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

(b)

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

(c)

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

(d)

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

(e)

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

(f)

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

(g)

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

(h)

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

(i)

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

(j)

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

(k)

$\left(\begin{array}{rr}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right)$ .

Problem 18.9

Using elementary operations, find the inverse of the matrix:

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(a)

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

(b)

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

(c)

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

(d)

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

(e)

$\left(\begin{array}{cccc}1 & 1 & \ldots & 1 \\ 0 & 1 & \ldots & 1 \\ \ldots & \ldots & \ldots & . \\ 0 & 0 & \ldots & 1\end{array}\right)$ ;

(f)

$\left(\begin{array}{cccccc}1 & 0 & 0 & \ldots & 0 & 0 \\ 1 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 & 1\end{array}\right)$ ;

(g)

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

(h)

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

(i)

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

(j)

$\left(\begin{array}{rrr}1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1\end{array}\right) ;$

(k)

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

Problem 18.10

(a)(b)

Find the inverse of the square matrix:

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(a)

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

(b)

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

Problem 18.11

(a)(b)

Find the inverse of the matrix:

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(a)

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

(b)

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

Problem 18.12

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

\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

(a)(b)

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 $A$ equal to, if the matrix $A^{-1}$ is also an integer matrix?

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Problem 18.15

(a)(b)

Let $A$ be a square matrix of order $n$ whose entries are polynomials in the variable $t$ , and let $\operatorname {det} A$ be a nonzero polynomial. Prove that there exists a unique matrix $B$ , whose entries are polynomials in $t$ , such that $A B=B A=(\operatorname {det} A) E$ . Find $B$ if:

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(a)

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

(b)

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

Problem 18.16

(a)(b)

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 $E+A B$ is invertible, then the matrix $E+B A$ is also invertible.

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Problem 18.18

Let $A$ and $B$ be matrices of sizes $n \times m$ and $m \times n$ respectively, such that $A B$ and $B A$ are identity matrices of sizes $n$ and $m$ . Prove that $m=n$ .

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Problem 18.19

Let $A$ be a matrix of size $m \times n$ having rank $m$ . Prove that there exists a matrix $X$ of size $n \times m$ such that $A X$ is the identity matrix of size $m$ .

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Problem 18.20

(a)(b)(c)(d)

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

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(a)

the $i$ -th and $j$ -th rows are swapped;

(b)

the $j$ -th row, multiplied by $c$ , is added to the $i$ -th row;

(c)

the $i$ -th row is multiplied by a number $c \neq 0$ ;

(d)

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

Problem 18.21

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

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Problem 18.22

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

\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 $B$ and $C$ be rows of length $n$ such that $C^{t} B \neq -1$ , and let $E$ be the identity matrix of size $n$ . Prove that the matrix $E+{ }^{t} B C$ is invertible.

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Problem 18.24

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

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