If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

3

(a)

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

3

(a)

If \(x\left[ \begin{matrix} 2 \\ 3 \end{matrix} \right] +y\left[ \begin{matrix} -1 \\ 1 \end{matrix} \right] =\left[ \begin{matrix} 10 \\ 5 \end{matrix} \right] \) , find the values of x and y.

3

(a)

Given:\(3\begin{bmatrix} x & y \\ z & w \end{bmatrix}=\begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix}+\begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix},\) find the values of x, y, z and w.

3

(a)

Given: \(3\begin{bmatrix} x & y \\ z & w \end{bmatrix}=\begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix}+\begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix},\)find the values of x,y,z and w.

3

(a)

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).

3

(a)

If \(A=diag.\left[ \begin{matrix} 3, & -5, & 7 \end{matrix} \right] \)then \(B=\left[ \begin{matrix} -1, & 2, & 4 \end{matrix} \right] ,\) then find \((2A+3B)\) .

3

(a)

Show that \(A+A\prime \) is symmetric when \(A=\begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix}\)

3

(a)

If \(A=\left[ \begin{matrix} 3 & 1 & -1 \\ 0 & 1 & 2 \end{matrix} \right] \) , then show that \(AA\prime \) is a symmetric matrix.

3

(a)

Show that: \(\left[ \left( \begin{matrix} 1 & \omega & { \omega }^{ 2 } \\ \omega & { \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & \omega \end{matrix} \right) +\left( \begin{matrix} \omega & { \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & \omega \\ \omega & { \omega }^{ 2 } & 1 \end{matrix} \right) \right] \left[ \begin{matrix} 1 \\ \omega \\ { \omega }^{ 2 } \end{matrix} \right] =\left[ \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right] ,\) where \(\omega \) is a cube root of unity.

3

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Using elementary transformations, find the inverse of matrix \(\begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}.\)

3

(a)

Using elementary transformations, find the inverse of matrix \(\begin{bmatrix} 4 & 5 \\ 3 & 4 \end{bmatrix}.\)

3

(a)

Using elementary transformations, find the inverse of matrix \(\begin{bmatrix} 2 & -6 \\ 1 & -2 \end{bmatrix}\) .

3

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Using elementary transformations, find the inverse of matrix \(\begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix}\)

3

(a)

If \(A=\begin{bmatrix} cos\theta & sin\theta \\ -sin\theta & cos\theta \end{bmatrix}\) , then prove that \({ A }^{ n }=\begin{bmatrix} cosn\theta & sinn\theta \\ -sinn\theta & cosn\theta \end{bmatrix}\)

3

(a)