Math Problems Vault

Problem 11.1

How does a determinant of order $n$ change if:

?

(a)

the sign of all its entries is reversed;

(b)

each entry $a_{i k}$ is multiplied by $c^{i-k}(c \neq 0)$ ;

(c)

each entry is replaced by the entry symmetric to it with respect to the anti-diagonal;

(d)

each entry is replaced by the entry symmetric to it with respect to the "center" of the determinant;

(e)

it is rotated by $90^{\circ }$ about its "center" (counterclockwise)?

Problem 11.2

(a)(b)

How does a determinant of order $n$ change if:

?

(a)

its first column is moved to the last position, and the remaining columns are shifted left, preserving their order;

(b)

its rows are written in reverse order?

Problem 11.3

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

How does the determinant change if:

?

(a)

the preceding column is added to each column, starting from the second;

(b)

all preceding columns are added to each column, starting from the second;

(c)

the following row is subtracted from each row except the last, and the original first row is subtracted from the last;

(d)

the preceding column is added to each column, starting from the second, and the original last column is added to the first?

Problem 11.4

Prove that the determinant of a skew-symmetric matrix of odd order equals 0.

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Problem 11.5

The numbers 20604, 53227, 25755, 20927, and 289 are divisible by 17. Prove that the determinant

\left[\begin{smallmatrix} 2 & 0 & 6 & 0 & 4 \\ 5 & 3 & 2 & 2 & 7 \\ 2 & 5 & 7 & 5 & 5 \\ 2 & 0 & 9 & 2 & 7 \\ 0 & 0 & 2 & 8 & 9 \end{smallmatrix}\right]

is also divisible by 17.

?

Problem 11.6

Compute, without expanding it, the determinant

\left[\begin{smallmatrix} x & y & z & 1 \\ y & z & x & 1 \\ z & x & y & 1 \\ \frac{x+z}{2} & \frac{x+y}{2} & \frac{y+z}{2} & 1 \end{smallmatrix}\right]

?

Problem 11.7

What is the value of a determinant in which the sum of the even-numbered rows equals the sum of the odd-numbered rows?

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Problem 11.8

Prove that any determinant equals half the sum of two determinants, one of which is obtained from the given one by adding the number $b$ to all entries of the $i$ -th row, and the other similarly by adding the number $-b$ .

?

Problem 11.9

Prove that if all entries of a determinant of order $n$ are differentiable functions of one variable, then the derivative of this determinant is the sum of $n$ determinants $D_{i}$ , where all rows of the determinant $D_{i}$ , except the $i$ -th, are the same as in the determinant $D$ , and the $i$ -th row consists of the derivatives of the entries of the $i$ -th row of the determinant $D$ .

?

Problem 11.10

Compute the determinants:

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(a)

$\left[\begin{smallmatrix} a_{1}+x & x & \ldots & x \\ x & a_{2}+x & \ldots & x \\ \ldots \ldots \ldots & \ldots \ldots & \ldots & \ldots \\ x & x & \ldots & a_{n}+x\end{smallmatrix}\right]$ ;

(b)

$\left[\begin{smallmatrix} a_{1}+x & a_{2} & \ldots & a_{n} \\ a_{1} & a_{2}+x & \ldots & a_{n} \\ \ldots \ldots \ldots & \ldots \ldots & \ldots & \ldots \\ a_{1} & a_{2} & \ldots & a_{n}+x\end{smallmatrix}\right]$ ;

(c)

$\left[\begin{smallmatrix} 1+x_{1} y_{1} & 1+x_{1} y_{2} & \ldots & 1+x_{1} y_{n} \\ 1+x_{2} y_{1} & 1+x_{2} y_{2} & \ldots & 1+x_{2} y_{n} \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots & \ldots \ldots & \ldots \\ 1+x_{n} y_{1} & 1+x_{n} y_{2} & \ldots & 1+x_{n} y_{n}\end{smallmatrix}\right]$ ;

(d)

$\left[\begin{smallmatrix} f_{1}\left(a_{1}\right) & f_{1}\left(a_{2}\right) & \ldots & f_{1}\left(a_{n}\right) \\ f_{2}\left(a_{1}\right) & f_{2}\left(a_{2}\right) & \ldots & f_{2}\left(a_{n}\right) \\ \ldots \ldots \ldots & \ldots \ldots & \ldots & \ldots \\ f_{n}\left(a_{1}\right) & f_{n}\left(a_{2}\right) & \ldots & f_{n}\left(a_{n}\right)\end{smallmatrix}\right]$ , where $f_{i}(x)$ is a polynomial of degree at most $n-2$ $(i=1,2, \ldots , n)$ ;

(e)

$\left[\begin{smallmatrix} 1+a_{1}+b_{1} & a_{1}+b_{1} & \ldots & a_{1}+b_{n} \\ a_{2}+b_{1} & 1+a_{2}+b_{2} & \ldots & a_{2}+b_{n} \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots & \ldots \ldots \ldots \\ a_{n}+b_{1} & a_{n}+b_{1} & \ldots & 1+a_{n}+b_{n}\end{smallmatrix}\right]$ ;

(f)

$\left[\begin{smallmatrix} 1+x_{1} y_{1} & x_{1} y_{2} & \ldots & x_{1} y_{n} \\ x_{2} y_{1} & 1+x_{2} y_{2} & \ldots & x_{2} y_{n} \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots & \ldots \ldots & \ldots \\ x_{n} y_{1} & x_{n} y_{2} & \ldots & 1+x_{n} y_{n}\end{smallmatrix}\right]$ .