If \(\left[ \begin{matrix} x & +3y & y \\ 7 & -x & 4 \end{matrix} \right] \)=\(\begin{bmatrix} 4 & -1 \\ 0 & 4 \end{bmatrix}\), find the values of x and y.

If \(\begin{bmatrix} 3 & 4 \\ 2 & x \end{bmatrix}\left[ \begin{matrix} x \\ 1 \end{matrix} \right] =\left[ \begin{matrix} 19 \\ 15 \end{matrix} \right] \), find the value of x

If A is a \(3\times 3\) matrix, whose elements are given by \({ a }_{ ij }=\frac { 1 }{ 3 } \left| -3i+j \right| \), then write the value \({ a }_{ 23 }\).

If \(A=\begin{bmatrix} 0 & a \\ 0 & 0 \end{bmatrix}\), find \({ A }^{ 16 }\).

If is \(A=\left[ \begin{matrix} 0 & b & -2 \\ 3 & 1 & 3 \\ 2a & 3 & -1 \end{matrix} \right] \)skew symmetric matrix, find the values of a and b.

Find \(\frac { 1 }{ 2 } \left( A+{ A }^{ \prime } \right) \) and \(\frac { 1 }{ 2 } \left( A-{ A }^{ \prime } \right) \) . If \(A=\left[ \begin{matrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{matrix} \right] \)

Let \(f(x)=\left[ \begin{matrix} cosx & -sinx & 0 \\ sinx & cosx & 0 \\ 0 & 0 & 1 \end{matrix} \right] \) Show that f(x) f(y) = f(x + y).

The bookshop of a particular school has 10 dozen Chemistry books, 8 dozen Physics books, 10 dozen Economics books.The selling prices are Rs. 80, Rs. 60 and Rs. 40 each respectively. Find the total amount, the bookshop will receive from selling all the books, using matrix algebra.

Express the following matrix as the sum of a symmetric and a skew symmetric matrix, and verify your result :

(i) \(\left[ \begin{matrix} 3 & -2 & -4 \\ 3 & -2 & -5 \\ -1 & 1 & 2 \end{matrix} \right] \)
(ii) \(\left[\begin{array}{rr} 3 & 5 \\ 1 & -1 \end{array}\right]\)
(iii) \(\left[\begin{array}{rrr} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{array}\right]\)
(iv) \(\left[\begin{array}{rr} 1 & 5 \\ -1 & 2 \end{array}\right]\)

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 \(A=\left( \begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix} \right) \) and A³ - 6A² + 7A + kI³ = 0, find k.