If X and Y are \(2\times 2\) matrices, then solve the following matrix equation of X and Y. 
\(2X+3Y=\begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix},3X+2Y=\begin{bmatrix} -2 & 2 \\ 1 & -5 \end{bmatrix}\)

4

(a)

If A is a square matrix such that A² = A, show that (l + A)³ = 7A + l

4

(a)

Let \(A=\begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}\), then show that \({ A }^{ 2 }-4A+7l=0\). Using this result, calculate \({ A }^{ 5 }\) also.

4

(a)

Find the matrix A, satisfying the matrix equation

\(\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}A\begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

4

(a)

Find the matrix X so that X \(\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix} \right] =\left[ \begin{matrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{matrix} \right] .\)

4

(a)

Express the matrix \(A=\left[\begin{array}{rrr}2 & 4 & -6 \\ 7 & 3 & 5 \\ 1 & -2 & 4\end{array}\right]\) as the sum of a symmetric and a skew-symmetric matrices.

4

(a)

 Express matrix A as the sum of a symmetric and a skew-symmetric matrices, where \(A=\left[\begin{array}{rrr}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]\)

4

(a)

If AB = BA for any two square matrices, then prove by mathematical induction that (AB) n = An B",

4

(a)