If for any 2 × 2 square matrix A, A(adj A) = $\left(\begin{array}{l l }8 & 0 \\ 0 & 8\end{array}\right)$ , then write the value of |A|.

Find the inverse of the matrix $\left(\begin{array}{l l }-3 & 2 \\ 5 & -3\end{array}\right)$ Hence, find the matrix P satisfying the matrix equation P$\left(\begin{array}{l l }-3 & 2 \\ 5 & -3\end{array}\right)$ = $\left(\begin{array}{l l }1 & 2 \\ 2 & -1\end{array}\right)$ .

If A and B are square matrices of order 3 such that |A| = –1, |B| = 3, then find the value of |2AB|.

Using properties of determinants, prove that : $\begin{vmatrix}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{vmatrix}$ = -2

If A = $\left(\begin{array}{l l }2 & 3 \\ 1 & -4\end{array}\right)$ , B = $\left(\begin{array}{l l }1 & -2 \\ -1 & 3\end{array}\right)$ , verify that $(A B)^{-1}=B^{-1}A^{-1}$

Using properties of determinants, solve for x : $\begin{vmatrix}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{vmatrix}$ = 0

Using properties of determinants, prove that : $\begin{vmatrix}1 & 1+p & 1+p+q \\ 3 & 4+3p & 2+4p+3p \\ 4 & 7+4p & 2+7p+4q\end{vmatrix}$ = 1

Without expanding the determinant at any stage; prove that : $\begin{vmatrix}0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0\end{vmatrix}$ = 0.

If A = $\left(\begin{array}{l l l }1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1\end{array}\right)$, find $\left(A^{\prime}\right)^{-1}.$

Using properties of determinants, prove the following : $\begin{vmatrix}a & a² & bc \\ b & b² & ca \\ c & c² & ab\end{vmatrix}$ = $(a-b)(b-c)(c-a)(b c+c a+a b).$

Using properties of determinants, prove that following : $\begin{vmatrix}a+x & y & z \\ x & a+y & z \\ x & y & a+z\end{vmatrix}$ = $a^{2}(a+x+y+z).$

Using properties of determinants, prove that $\begin{vmatrix}1 & 1 & 1+3x \\ 1+3y & 1 & 1 \\ 1 & 1+3z & 1\end{vmatrix}$ = $9\bigl(3x y z+x y+y z+z x\bigr)$

Using properties of determinants, prove that $\begin{vmatrix}x+λ & 2x & 2x \\ 2x & x+λ & 2x \\ 2x & 2x & x+λ\end{vmatrix}$ = $(5x+{\lambda})({\lambda}-x)^{2}.$

Prove the following : $\begin{vmatrix}a² & bc & ac+c² \\ a²+ab & b² & ac \\ ab & b²+bc & c²\end{vmatrix}$ = $4a^{2}b^{2}c^{2}.$

Using properties of determinants, prove that : $\begin{vmatrix}2y & y-z-x & 2y \\ 2z & 2z & z-x-y \\ x-y-z & 2x & 2x\end{vmatrix}$ = $(x+y+z)^{3}.$