For any 2 \(\times\) 2 matrix, if A (adj A) =\(\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right] \) then find |A|.

For the matrix A, if A³ = I, then find A^-1.

If A is a square matrix such that A³ = I, then prove that A is non-singular.

Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

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

Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.

Find k if the equations x + 2y + 2z = 0, x - 3y - 3z = 0, 2x + y + kz = 0 have only the trivial solution.

Solve : 2x - y = 3, 5x + y = 4 using matrices.

Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

Use matrices to find the solution set of
x + y - 2z = 3
3x - y + z = 5
3x + 3y - 6z = 9

If A  \(=\left[\begin{array}{lll} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1 \end{array}\right]\), \(A^{-1}=\left[\begin{array}{ccc} \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\ -4 & 3 & c \\ \frac{5}{2} & -\frac{3}{2} & \frac{1}{2} \end{array}\right]\) then find a and c.

If A \(=\left[\begin{array}{ccc} -1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{array}\right]\)then find adj A.

If A  \(=\left[\begin{array}{ccc} 1 & 2 & 1 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{array}\right]\)then find A³

If \(A=\left[\begin{array}{ll} 3 & -4 \\ 1 & -1 \end{array}\right]\), then find A².