Math Problems Vault

Problem 16.1

(a)(b)

Find the greatest possible value of a third-order determinant composed:

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(a)

of the numbers 0 and 1;

(b)

of the numbers 1 and -1.

Problem 16.2

Prove that if, in a determinant of order $n$ , the entries at the intersection of some $k$ rows and $l$ columns are all zero, and $k+l>n$ , then the determinant equals zero.

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Problem 16.3

Let $D$ be a determinant of order $n>1$ , and let $D_{1}$ and $D_{2}$ be the determinants obtained from $D$ by replacing each entry $a_{i j}$ by its cofactor $A_{i j}$ for $D_{1}$ , and by its minor $M_{i j}$ for $D_{2}$ . Prove that $D_{1}=D_{2}$ .

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Problem 16.4

(a)(b)

The adjugate (or adjoint) matrix $\widehat{A}$ of a square matrix $A$ of order $n$ is the matrix whose $(i,j)$ entry is the cofactor $A_{j i}$ . Prove that:

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(a)

$\left|\widehat{A}\right|=\left|A\right|^{n-1}$ ;

(b)

$\hat{\hat{A}}=\left|A\right|^{n-2} A$ for $n>2$ , and $\hat{\hat{A}}=A$ for $n=2$ .

Problem 16.5

Let $A=\left(a_{i j}\right), B=\left(b_{i j}\right)$ be matrices of size $m \times n$ , and let $A_{i_{1}, \ldots , i_{m}}$ and $B_{i_{1}, \ldots , i_{m}}$ be the minors of order $m$ of the matrices $A$ and $B$ , respectively, formed from the columns numbered $i_{1}, \ldots , i_{m}$ ,

c_{i j}=\sum _{k=1}^{n} a_{i k} b_{j k}, \quad C=\left(c_{i j}\right), \quad i=1, \ldots , m ; \quad j=1, \ldots , m

Prove that

\operatorname {det} C=\sum _{1 \leqslant i_{1}<i_{2}<\ldots <i_{m} \leqslant n} A_{i_{1}, \ldots , i_{m}} B_{i_{1}, \ldots , i_{m}}

if $m \leqslant n$ , and $\operatorname {det} C=0$ if $m>n$ .

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Problem 16.6

Let $A$ and $B$ be matrices of sizes $p \times n$ and $n \times k$ , respectively, $A\left(\begin{array}{lll}i_{1} & \ldots & i_{m} \\ j_{1} & \ldots & j_{m}\end{array}\right), \quad B\left(\begin{array}{lll}i_{1} & \ldots & i_{m} \\ j_{1} & \ldots & j_{m}\end{array}\right)$ the minors of the matrices $A$ and $B$ lying at the intersections of the rows numbered $i_{1}, \ldots , i_{m}$ and the columns numbered $j_{1}, \ldots , j_{m}$ , and let $C=A B$ . Prove that

C\left(\begin{array}{lll}i_{1} & \ldots & i_{m} \\ j_{1} & \ldots & j_{m}\end{array}\right) = \begin{cases} \displaystyle \sum _{1 \leqslant k_{1}<k_{2}<\ldots <k_{m} \leqslant n} A\left(\begin{array}{lll}i_{1} & \ldots & i_{m} \\ k_{1} & \ldots & k_{m}\end{array}\right) B\left(\begin{array}{lll}k_{1} & \ldots & k_{m} \\ j_{1} & \ldots & j_{m}\end{array}\right), & \text{ if } m \leqslant n, \\ 0, & \text{ if } m>n. \end{cases}

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Problem 16.7

Prove that the sum of the principal minors of order $k$ of the matrix $A \cdot { }^{t} A$ equals the sum of the squares of all minors of order $k$ of the matrix $A$ .

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Problem 16.8

Let

D=\left[\begin{smallmatrix} a_{11} & \ldots & a_{1 n} \\ \ldots \ldots & \ldots & \ldots \\ a_{n 1} & \ldots & a_{n n} \end{smallmatrix}\right]

Prove that

\left[\begin{smallmatrix} a_{11} & \ldots & a_{1 n} & x_{1} \\ \ldots \ldots & \ldots & \ldots & \ldots \\ a_{n 1} & \ldots & a_{n n} & x_{n} \\ x_{1} & \ldots & x_{n} & z \end{smallmatrix}\right]=D z-\sum _{i, j=1}^{n} A_{i j} x_{i} x_{j}

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Problem 16.9

Prove that the sum of the cofactors of the entries of a row of a determinant does not change if the same number is added to all entries of the matrix.

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Problem 16.10

Prove that if all entries of some row (column) of a determinant equal 1, then the sum of the cofactors of all entries of the determinant equals the determinant itself.

* \quad * \quad *

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Problem 16.11

Let

A=\left[\begin{smallmatrix} a_{11} & \ldots & a_{1 n} \\ \ldots \ldots \ldots & \ldots & \ldots \\ a_{n 1} & \ldots & a_{n n} \end{smallmatrix}\right], \quad B=\left[\begin{smallmatrix} b_{11} & \ldots & b_{1 k} \\ \ldots \ldots & \ldots & \ldots \\ b_{k 1} & \ldots & b_{k k} \end{smallmatrix}\right],

D=\left\lvert \, \begin{array}{llllllllll} a_{11} b_{11} & \ldots & a_{1 n} b_{11} & a_{11} b_{12} & \ldots & a_{1 n} b_{12} & \ldots & a_{11} b_{1 k} & \ldots & a_{1 n} b_{1 k} \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ a_{n 1} b_{11} & \ldots & a_{n n} b_{11} & a_{n 1} b_{12} & \ldots & a_{n n} b_{12} & \ldots & a_{n 1} b_{1 k} & \ldots & a_{n n} b_{1 k} \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \end{array}.\right.

where $D$ (a determinant of order $n k$ ) is the Kronecker product of the determinants $A$ and $B$ . Prove that $D=A^{k} B^{n}$ .

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Problem 16.12

(a)(b)(c)

A continuant is the determinant

\left(a_{1} a_{2} \ldots a_{n}\right)=\left[\begin{smallmatrix} a_{1} & 1 & 0 & 0 & \ldots & 0 & 0 \\ -1 & a_{2} & 1 & 0 & \ldots & 0 & 0 \\ 0 & -1 & a_{3} & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & -1 & a_{n} \end{smallmatrix}\right].

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(a)

Express $(a_{1} a_{2} \ldots a_{n})$ as a polynomial in $a_{1}, \ldots , a_{n}$ .

(b)

Write the expansion of the continuant along the first $k$ rows.

(c)

Establish the following connection of the continuant with continued fractions:

\frac{\left(a_{1} a_{2} \ldots a_{n}\right)}{\left(a_{2} a_{3} \ldots a_{n}\right)}=a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\ldots +\frac{1}{\ddots } \frac{1}{a_{n-1}+\frac{1}{a_{n}}}}}

Problem 16.13

Prove that if $A, B, C, D$ are square matrices of order $n$ and $C \cdot { }^{t} D=D \cdot { }^{t} C$ , then

\left[\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}\right]=\left|A \cdot { }^{t} D-B \cdot { }^{t} C\right|

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Problem 16.14

Prove that if $A, B, C, D$ are square matrices of order $n$ , where $C$ or $D$ is a nonsingular matrix, and $C D=D C$ , then

\left[\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}\right]=\left|A D-B C\right|

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Problem 16.15

Compute the determinant $\left|\begin{array}{cc}c E & A \\ A & c E\end{array}\right|$ , where

A=\left[\begin{smallmatrix} a & 1 & 0 & 0 & \ldots & 0 \\ 1 & a & 1 & 0 & \ldots & 0 \\ 0 & 1 & a & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & a \end{smallmatrix}\right]

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Problem 16.16

Prove that the determinant obtained by deleting the $k$ -th column of the matrix

\left(a_{i j}\right)=\left(\binom {j-1}{i-1}\right), \quad i=1, \ldots , n+1 ; \quad j=1, \ldots , n+2

equals $\binom {n+1}{k-1}$ .

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Problem 16.17

Prove that

\left[\begin{smallmatrix} \frac{1}{2!} & \frac{1}{3!} & \frac{1}{4!} & \cdots & \frac{1}{(2 k+2)!} \\ 1 & \frac{1}{2!} & \frac{1}{3!} & \cdots & \frac{1}{(2 k+1)!} \\ 0 & 1 & \frac{1}{2!} & \cdots & \frac{1}{(2 k)!} \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & \frac{1}{2!} \end{smallmatrix}\right]=0, \quad k \in \mathbb {N}.

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Problem 16.18

Multiplying the matrices

\left[\begin{smallmatrix} x_{1} & x_{2} & x_{3} & x_{4} \\ x_{2} & -x_{1} & -x_{4} & x_{3} \\ x_{3} & x_{4} & -x_{1} & -x_{2} \\ x_{4} & -x_{3} & x_{2} & -x_{1} \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} y_{1} & y_{2} & y_{3} & y_{4} \\ y_{2} & -y_{1} & -y_{4} & y_{3} \\ y_{3} & y_{4} & -y_{1} & -y_{2} \\ y_{4} & -y_{3} & y_{2} & -y_{1} \end{smallmatrix}\right],

prove that

\begin{aligned} & \left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}\right)\left(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}\right)= \\ & \quad =\left(x_{1} y_{1}+x_{2} y_{2}+x_{3} y_{3}+x_{4} y_{4}\right)^{2}+\left(x_{1} y_{2}-x_{2} y_{1}-x_{3} y_{4}+x_{4} y_{3}\right)^{2}+ \\ & \quad +\left(x_{1} y_{3}+x_{2} y_{4}-x_{3} y_{1}-x_{4} y_{2}\right)^{2}+\left(x_{1} y_{4}-x_{2} y_{3}+x_{3} y_{2}-x_{4} y_{1}\right)^{2} \end{aligned}

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Problem 16.19

(a)(b)

Compute the determinant of the matrix $(a_{i j})$ of order $n$ , where:

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(a)

$a_{i j}$ equals 1 if $i$ divides $j$ , and equals 0 otherwise;

(b)

$a_{i j}$ equals the number of common divisors of $i$ and $j$ .

Problem 16.20

Prove that the determinant of the matrix $(d_{i j})$ of order $n$ , where $d_{i j}$ is the greatest common divisor of $i$ and $j$ , equals $\varphi (1) \varphi (2) \ldots \varphi (n)$ .

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Problem 16.21

Let $x_{1} \ldots x_{n}, y_{1} \ldots y_{n}$ be numbers such that $x_{i} y_{j} \neq 1$ for all $i, j=1, \ldots , n, \quad \Delta \left(x_{1}, \ldots , x_{n}\right), \Delta \left(y_{1}, \ldots , y_{n}\right)$ are Vandermonde determinants. Prove that

\Delta \left(x_{1}, \ldots , x_{n}\right) \Delta \left(y_{1}, \ldots , y_{n}\right)=\operatorname {det}\left(\frac{1}{1-x_{i} y_{j}}\right)_{i, j=1, \ldots , n} \prod _{i, j=1}^{n}\left(1-x_{i} y_{j}\right)

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