Math Problems Vault

Problem I.3.2.1

The integers $1798,2139,3255,4867$ are divisible by 31. Without any computation, show that the fourth-order determinant

\begin{vmatrix} 1 & 7 & 9 & 8 \\ 2 & 1 & 3 & 9 \\ 3 & 2 & 5 & 5 \\ 4 & 8 & 6 & 7 \end{vmatrix}

is also divisible by 31.

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Problem I.3.2.2

Show that any skew-symmetric fourth-order determinant $\left|a_{i j}\right|$ with $a_{i j} \in \mathbb {Z}$ is the square of an integer.

Remark. This is true for a skew-symmetric determinant of arbitrary order.

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Problem I.3.2.3

Prove the relation $\operatorname {det} A B=\operatorname {det} A \cdot \operatorname {det} B$ (Theorem 3 $)$ by reducing, via elementary transformations of type (II) on the rows, the auxiliary matrix $C=\left\Vert \begin{smallmatrix} E & B \\ -A & 0\end{smallmatrix}\right\Vert$ of size $2 n \times 2 n$ to the form $C^{\prime }=\left\Vert \begin{smallmatrix} E & B \\ 0 & A B\end{smallmatrix}\right\Vert$ .

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Problem I.3.2.4

In problems on the modeling of random stationary processes, determinants of the form

\Delta _{n}\left(k_{1}, x_{1} ; \ldots ; k_{m}, x_{m}\right)=\begin{vmatrix} M_{k_{1}}^{n}\left(x_{1}\right) \\ M_{k_{2}}^{n}\left(x_{2}\right) \\ \ldots \ldots \ldots \\ M_{k_{m}}^{n}\left(x_{m}\right) \end{vmatrix}

arise, where $x_{1}, \ldots , x_{m}$ are arbitrary variables; $k_{1}, \ldots , k_{m}$ are natural numbers, $k_{1}+k_{2}+\ldots +k_{m}=n$ ; $M_{k}^{n}(x)$ is a $k \times n$ matrix of the form

M_{k}^{n}(x)=\left\Vert \begin{smallmatrix} 1 & x & x^{2} & \ldots & x^{n-1} \\ 0 & 1 & \binom {2}{1}x & \ldots & \binom {n-1}{1}x^{n-2} \\ 0 & 0 & 1 & \ldots & \binom {n-1}{2}x^{n-3} \\ \ldots & \ldots & \ldots \ldots \ldots \ldots \\ 0 & 0 & 0 & \ldots & \binom {n-1}{k-1}x^{n-k} \end{smallmatrix}\right\Vert .

Prove that

\Delta _{n}\left(k_{1}, x_{1} ; \ldots ; k_{m}, x_{m}\right)=\prod _{1 \leqslant j<i \leqslant m}\left(x_{i}-x_{j}\right)^{k_{i} k_{j}}

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Problem I.3.2.5

Show that

\begin{aligned} B_n(s,t) = \begin{vmatrix} \binom {s}{t} & \binom {s}{t+1} & \cdots & \binom {s}{t+n-1} \\ \binom {s+1}{t} & \binom {s+1}{t+1} & \cdots & \binom {s+1}{t+n-1} \\ \vdots & \vdots & \vdots & \vdots \\ \binom {s+n-1}{t} & \binom {s+n-1}{t+1} & \cdots & \binom {s+n-1}{t+n-1} \end{vmatrix} \\ & =\frac{\binom {n+s-1}{n}\binom {n+s-2}{n} \cdots \binom {n+s-t}{n}}{\binom {n+t-1}{n}\binom {n+t-2}{n} \cdots \binom {n}{n}}. \end{aligned}

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Problem I.3.2.6

Let

C_{n}\left(\lambda _{1}, \ldots , \lambda _{n}\right)=\left\Vert \begin{smallmatrix} \lambda _{1} & 1 & 0 & \ldots & 0 & 0 & 0 \\ -1 & \lambda _{2} & 1 & \ldots & 0 & 0 & 0 \\ \ldots & \ldots \ldots & \ldots \ldots \ldots & \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\ 0 & 0 & 0 & \ldots & \lambda _{n-2} & 1 & 0 \\ 0 & 0 & 0 & \ldots & -1 & \lambda _{n-1} & 1 \\ 0 & 0 & 0 & \ldots & 0 & -1 & \lambda _{n} \end{smallmatrix}\right\Vert .

Show that $\operatorname {det} C_{n}=\lambda_{n} \operatorname {det} C_{n-1}+\operatorname {det} C_{n-2}$ . For $\lambda_{1}=\lambda_{2}=\ldots =\lambda_{n}=1$ , find the numerical value of $\operatorname {det} C_{n}$ .

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Problem I.3.2.7

Show that the determinant of the $n \times n$ matrix

A_{n}=\left\Vert \begin{smallmatrix} 2 & -1 & 0 & 0 & \ldots & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & \ldots & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & \ldots & 0 & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots . \\ 0 & 0 & 0 & 0 & \ldots & -1 & 2 & -1 \\ 0 & 0 & 0 & 0 & \ldots & 0 & -1 & 2 \end{smallmatrix}\right\Vert

equals $n+1$ .

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Problem I.3.2.8

Let $A, B$ be any square matrices of order $n$ . Show that

\operatorname {det}\left\Vert \begin{smallmatrix} A & B \\ B & A \end{smallmatrix}\right\Vert =\operatorname {det}(A+B) \cdot \operatorname {det}(A-B)

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Problem I.3.2.9

Let $X$ be a matrix of size $n \times k$ , and $Y$ of size $k \times n$ . Prove that

\operatorname {det}\left(E_{n}+X Y\right)=\operatorname {det}\left(E_{k}+Y X\right)

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