Math Problems Vault

Problem 10.1

(a)(b)(c)

Determine which of the following products occur in the expanded expression of the determinants of the corresponding order, and with what signs:

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(a)

$a_{13} a_{22} a_{31} a_{46} a_{55} a_{64}$ ;

(b)

$a_{31} a_{13} a_{52} a_{45} a_{24}$ ;

(c)

$a_{34} a_{21} a_{46} a_{17} a_{73} a_{54} a_{62}$ .

Problem 10.2

Choose the values of $i, j, k$ so that the product

a_{51} a_{i 6} a_{1 j} a_{35} a_{44} a_{6 k}

occurs in the expanded expression of the sixth-order determinant with a minus sign.

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Problem 10.3

In the expanded expression of the determinant

\left[\begin{smallmatrix} x & 1 & 2 & 3 \\ x & x & 1 & 2 \\ 1 & 2 & x & 3 \\ x & 1 & 2 & 2 x \end{smallmatrix}\right]

find the terms containing $x^{4}$ and $x^{3}$ .

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Problem 10.4

Using the definition, compute the following determinants:

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(a)

$\left[\begin{smallmatrix} a_{11} & 0 & 0 & \ldots & 0 \\ a_{21} & a_{22} & 0 & \ldots & 0 \\ a_{31} & a_{32} & a_{33} & \ldots & 0 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots \\ a_{n 1} & a_{n 2} & a_{n 3} & \ldots & a_{n n}\end{smallmatrix}\right] ;$

(b)

$\left[\begin{smallmatrix} 0 & \ldots & 0 & 0 & a_{1 n} \\ 0 & \ldots & 0 & a_{2, n-1} & a_{2 n} \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots \\ a_{n 1} & \ldots & a_{n, n-2} & a_{n, n-1} & a_{n n}\end{smallmatrix}\right]$ ;

(c)

$\left|\begin{array}{llll}a & 3 & 0 & 5 \\ 0 & b & 0 & 2 \\ 1 & 2 & c & 3 \\ 0 & 0 & 0 & d\end{array}\right|$ ;

(d)

$\left|\begin{array}{llll}1 & 0 & 2 & a \\ 2 & 0 & b & 0 \\ 3 & c & 4 & 5 \\ d & 0 & 0 & 0\end{array}\right|$ ;

(e)

$\left[\begin{smallmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & 0 & 0 & 0 \\ a_{41} & a_{42} & 0 & 0 & 0 \\ a_{51} & a_{52} & 0 & 0 & 0\end{smallmatrix}\right]$ .

Problem 10.5

Express, as a polynomial arranged in decreasing powers of $t$ , the determinant

\left[\begin{smallmatrix} -t & 0 & 0 & \ldots & 0 & a_{1} \\ a_{2} & -t & 0 & \ldots & 0 & 0 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & -t & 0 \\ 0 & 0 & 0 & \ldots & a_{n} & -t \end{smallmatrix}\right].

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Problem 10.6

Compute the determinant in which all entries of the main diagonal equal 1, the entries of the column numbered $j$ equal $a_{1}, a_{2}, \ldots , a_{j-1}, a_{j+1}, \ldots , a_{n}$ , and all remaining entries equal 0.

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Problem 10.7

Let

\sigma =\left[\begin{smallmatrix} 1 & 2 & \ldots & n \\ i_{1} & i_{2} & \ldots & i_{n} \end{smallmatrix}\right] \in \mathbf{S}_{n}

and let $A$ be a square matrix of order $n$ with entries $a_{r s}$ , where $a_{r s}=1$ if $s=i_{r}$ , and $a_{r s}=0$ otherwise. Prove that the determinant of the matrix $A$ equals the sign of the permutation $\sigma$ .

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