If |adj(adj A)| = |A|⁹, then the order of the square matrix A is

If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T = 

The system of linear equations x + y + z  = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________

If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________

The rank of any 3 \(\times\) 4 matrix is
(1) May be 1
(2) May be 2
(3) May be 3
(4) Maybe 4

If A is symmetric then
(1) A^T = A
(2) adj A is symmetric
(3) adj (A^T) = (adj A)^T
(4) A is orthogonal

If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.

If A is a non-singular matrix of odd order, prove that |adj A| is positive

Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.

Find the rank of the matrix \(\left[ \begin{matrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{matrix} \right] \).

Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).

Find a matrix A if adj(A) = \(\left[ \begin{matrix} 7 & 7 & -7 \\ -1 & 11 & 7 \\ 11 & 5 & 7 \end{matrix} \right] \).

Verify the property (A^T)^-1 = (A^-1)^T with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).

Solve: 3x+ay = 4, 2x + ay = 2, a ≠ 0 by Cramer's rule.

Verify (AB)^-1 = B^-1 A^-1 for A =\(\left[ \begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix} \right] \) and B =\(\left[ \begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix} \right] \).

If A = \(\left[ \begin{matrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{matrix} \right] \), verify thatA(adj A) = (adj A)A = |A| I₃.