Construct a 2 \(\times\) 3 matrix whose (i, j)^th element is given by \(a_ij={\sqrt{3}\over 2}|2i-3j|(1\le i\le2,1\le j\le3)\) .

Solve for x if \(\left[\begin{array}{lll} x & 2 & -1 \end{array}\right]\)\(\begin{bmatrix} 1&1 &2 \\ -1 & -4 &1 \\ -1 &-1 &-2 \end{bmatrix}\)\(\begin{bmatrix} x \\ 2 \\ 1 \end{bmatrix}\)=0

Consider the matrix Aa=\(\begin{bmatrix} cos \alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\)
Find all possible real values of α satisfying the condition \(A\alpha +A^T_{\alpha}=I\)

If A =\(\begin{bmatrix} 1 &0 &0 \\0 & 1 & 0 \\a &b &-1 \end{bmatrix}\) , show that A² is a unit matrix.

Verify the property A(B + C) = AB + AC, when the matrices A, B, and C are given by
A =\(\begin{bmatrix} 2 & 0&-3 \\1 & 4&5 \end{bmatrix}\),B =\(\begin{bmatrix} 3 & 1 \\ -1 & 0 \\ 4 & 2 \end{bmatrix}\) and C =\(\begin{bmatrix} 4 & 7 \\ 2 & 1 \\ 1 & -1 \end{bmatrix}\).

For what value of x, the matrix A = \(\begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & x^3 \\ 2 & -3 & 0 \end{bmatrix}\) is skew-symmetric.

If \(\begin{bmatrix} 0 & p& 3 \\ 2 & q^2 & -1 \\ r & 1 & 0 \end{bmatrix}\) is skew-symmetric, find the values of p, q, and r.

If A = \(\begin{bmatrix} {1\over 2}&\alpha \\ 0 & {1\over2} \end{bmatrix}\) ,prove that \(\sum^n_{k=1}det(A^k)={1\over3}(1-{1\over 4^n}).\)

If cos 2 \(\theta\) = 0 , determine\(\begin{vmatrix} 0& cos \theta &sin \theta \\ cos \theta &sin \theta &0 \\ sin \theta & 0 & cos \theta \end{vmatrix}^2\).