Suppose that a matrix has 12 elements. What are the possible orders it can have? What if it has 7 elements?

Find x, y, a, and b if \(\begin{bmatrix} 3x+4y & 6 & x-2y \\ a+b & 2a-b & -3 \end{bmatrix}\)=\(\begin{bmatrix} 2 & 6 & 4 \\ 5 & -5 & -3 \end{bmatrix}\)

Compute A + B and A - B if A =\(\begin{bmatrix} 4 & \sqrt { 5 } & 7 \\ -1 & 0 & 0.5 \end{bmatrix}\) and B = \(\begin{bmatrix} \sqrt { 3 } & \sqrt { 5 } & 7.3 \\ 1 & {1\over3} &{1\over4} \end{bmatrix}\) .

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\) verify (A - B)^T = A^T - B^T

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\)
verify (3A)^T = 3A^T

Construct an m \(\times\) n matrix A = [a_ij], where a _ij is given by
\(a_{ij}={(i-2j)^2\over 2}with \ m=2,n=3\)

Determine the value of x + y if \(\begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix}=\begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix}\)

Give your own examples of matrices satisfying the following conditions in each case:
(i) A and B such that AB \(\neq\) BA.
(ii) A and B such that \(A B=O=B A, A \neq O \text {and } B \neq O \text {. }\)
(iii) A and B such that \(A B=O \text {and } B A \neq O\)

If A^T=\(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B =  \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\), verify (B^T)^T = B

Evaluate :\(\begin{vmatrix} cos \theta & sin \theta \\ -sin \theta & cos \theta \end{vmatrix}\)