91Ó°ÊÓ

If \(A\) and \(B\) be symmetric matrices of the same order, then show that i. \(A+B\) is a symmetric matrix ii. \(A B-B A\) is a skew-symmetric matrix iii. \(A B+B A\) is a symmetric matrix

If \(A=\left[\begin{array}{ccc}1 & -2 & 3 \\ 2 & 3 & -1 \\ -3 & 1 & 2\end{array}\right]\), find det \(A .\) Ans. 42\(\\}\)

For what value of \(k,\left[\begin{array}{ccc}1 & 2 & -3 \\ 5 & 0 & k \\ 1 & -1 & 1\end{array}\right]\) is a singular matrix.

If \(A\) and \(B\) are two square matrices of order 3 such that \(|A|=-1,|B|=3\), then find \(|3 A B|\).

Prove that a skew-symmetric matrix of odd order must be a singular matrix.

If \(A=\left[\begin{array}{cc}-5 & 2 \\ 1 & -3\end{array}\right]\), then find \(\operatorname{adj} A\).

Find the adjoint of \(\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3\end{array}\right]\).

Find the adjoint of the matrix \(A=\left[\begin{array}{ccc}1 & 0 & -1 \\ 3 & 4 & 5 \\ 0 & -6 & -7\end{array}\right]\) and verify that i. \(\quad A(a d j A)=|A| I_{3}=(a d j A) A\) ii. \(\quad|a d j A|=|A|^{2}\) iii. \(a d j A^{T}=(a d j A)^{T}\) iv. \(\operatorname{adj}(\operatorname{adj} A)=|A| A\).

If \(A=\left[\begin{array}{ccc}1 & -2 & 3 \\ 2 & 3 & -1 \\ -3 & 1 & 2\end{array}\right]\) and \(B=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 1 & 2 \\\ 1 & 2 & 0\end{array}\right]\), then verify that \(\operatorname{adj} A B=(\operatorname{adjB})(\) adj \(A)\).

If \(A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\) and \(A(a d j A)=k\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]\), then find the value of \(k .\\{\) Ans. 1\(\\}\)