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 = 

If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) = 

If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A = 

If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| = 

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

If adj A = \(\left[ \begin{matrix} -1 & 2 & 2 \\ 1 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right] \), find A⁻¹.

If A is symmetric, prove that then adj A is also symmetric.

Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal.

Find the rank of the following matrices by row reduction method:
\(\left[ \begin{matrix} 1 \\ \begin{matrix} 3 \\ \begin{matrix} 1 \\ 1 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} -1 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} -1 \\ \begin{matrix} 2 \\ \begin{matrix} 3 \\ 1 \end{matrix} \end{matrix} \end{matrix} \right] \)

Find the rank of the following matrices by row reduction method:
\(\left[ \begin{matrix} 3 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} -8 \\ \begin{matrix} -5 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 5 \\ \begin{matrix} 1 \\ 3 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix} \right] \)

Solve the following system of linear equations by matrix inversion method :
2x  −  y  =  8 ,   3x  +  2y  =  −2.

Solve the following systems of linear equations by Cramer’s rule:
 \(\frac { 3 }{ x } \) + 2y = 12, \(\frac { 2 }{ x } \) + 3y = 13

Find the rank of each of the following matrices:
\(\left[ \begin{matrix} 4 & 3 \\ -3 & -1 \\ 6 & 7 \end{matrix}\begin{matrix} 1 & -2 \\ -2 & 4 \\ -1 & 2 \end{matrix} \right] \)

Solve the following system of linear equations by matrix inversion method:
2x + 3y − z = 9, x + y + z = 9, 3x − y − z  = −1

Solve the following system of linear equations by matrix inversion method:
x + y + z − 2 = 0, 6x − 4y + 5z − 31 = 0, 5x + 2y + 2z = 13.

Solve the following systems of linear equations by Cramer’s rule:
 3x + 3y − z = 11, 2x − y + 2z = 9, 4x + 3y + 2z = 25.

Solve the following systems of linear equations by Cramer’s rule:
\(\frac { 3 }{ x } -\frac { 4 }{ y } -\frac { 2 }{ z } \) -1 = 0, \(\frac { 1 }{ x } +\frac { 2 }{ y } +\frac { 1 }{ z } \) - 2 = 0, \(\frac { 2 }{ x } -\frac { 5 }{ y } -\frac { 4 }{ z } \) + 1 = 0