Using elementary transformations, find the inverse of the matrix

\(\left[ \begin{matrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{matrix} \right] \).

If \(A=\left[ \begin{matrix} \cos { \alpha } & \sin { \alpha } \\ -\sin { \alpha } & \cos { \alpha } \end{matrix} \right] ,\) then show that \({ A }^{ 2 }=\begin{bmatrix} \cos { 2\alpha } & \sin { 2\alpha } \\ -\sin { 2\alpha } & \cos { 2\alpha } \end{bmatrix}\)

If \(A=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\) and \(l=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\), prove that \((al+bA)^{ 3 }={ a }^{ 3 }l+{ 3a }^{ 2 }bA.\)

Let f(x) = x² - 5x+6 find f(A) If, A =\(\left[ \begin{matrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{matrix} \right] \)

If, A = \(\left[ \begin{matrix} a & 0 \\ 1 & 1 \end{matrix} \right] \)and B = \(\left[ \begin{matrix} 1 & 0 \\ 5 & 1 \end{matrix} \right] \) find all those values of a for which A = B

If \(A=\left( \begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix} \right) \), prove that A³ - 6A² + 7A + 2I = 0

If \(A=\left( \begin{matrix} 1 & 3 & 2 \\ 2 & 0 & -1 \\ 1 & 2 & 3 \end{matrix} \right) \), then show that A³ - 4A² - 3A + 11I = 0

Using elementary column operations, find the inverse of the following matrix :
\(\left[ \begin{matrix} -1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix} \right] \)

If  \(A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right] \) Prove that , A =\(\left[ \begin{matrix} { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \\ { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \\ { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \end{matrix} \right] \) for every positive integer n.