行列式：练习题

Find the determinant of the matrix \(A = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}\).

Let \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) be a matrix such that its determinant \( \det(A) = 5 \). Find the determinant of the matrix \( M = A^2 - (a+d)A \).

Let \(A\) and \(B\) be \(3 \times 3\) matrices. If \(\det(A) = 3\) and \(\det(B) = -2\), find the value of \(\det(A^2 B)\).

Find the value of \(x\) for which the matrix \(M = \begin{pmatrix} x+1 & 3 \\ 4 & x-2 \end{pmatrix}\) is singular. Then, if \(N = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}\), find \(\det(MN)\) for this value of \(x\).

Given the matrix \(A = \begin{pmatrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{pmatrix}\), find all possible real values of \(k\) such that the matrix \(A\) is singular.

Let \( M = \begin{pmatrix} 1 & k & 2 \\ 0 & 3 & -1 \\ 2 & 1 & 0 \end{pmatrix} \), where \( k \) is a constant.
(a) Find the determinant of \( M \) in terms of \( k \).
(b) Given that the determinant of \( M \) is \( 7 \), find the value of \( k \).

Consider the matrix \(M = \begin{pmatrix} 1 & 2 & k \\ 3 & 1 & 0 \\ 0 & 4 & -1 \end{pmatrix}\), where \(k\) is a real number.
(a) Find the determinant of \(M\) in terms of \(k\).
(b) Find the value(s) of \(k\) for which the matrix \(M\) is singular.
(c) For the largest value of \(k\) found in part (b), determine the cofactor of the entry in the first row and third column, \(C_{13}\).