Chapter 10

Prove that the matrix \(A=\left[\begin{array}{ccc}-5 & -8 & 0 \\ 3 & 3 & 0 \\\ 1 & 2 & -1\end{array}\right]\) is involutary.

Determine the condition that the matrix \(A=\left[\begin{array}{cc}a & b \\ c & -a\end{array}\right]\) is involutary.

Prove that the matrix \(A=\left[\begin{array}{ccc}1 & -2 & -6 \\ -3 & 1 & 9 \\\ 2 & 0 & -3\end{array}\right]\) is periodic whose period is 2 .

If \(A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0\end{array}\right]\), find \(A^{2}\) and show that \(A^{2}=A^{-1}\). Is \(A\) a periodic matrix? If yes, find its period.

If \(A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0\end{array}\right]\), find \(A^{2}\) and show that \(A^{2}=A^{-1}\). Is \(A\) a periodic matrix? If yes, find its period.

Show that the matrix \(A=\left[\begin{array}{cc}a b & b^{2} \\ -a^{2} & -a b\end{array}\right]\) is nilpotent of index \(2 .\)

Show that the matrix \(A=\left[\begin{array}{ccc}1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3\end{array}\right]\) is nilpotent of index 3 .

Find the rank of the matrix \(A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\\ 2 & 2 & 1\end{array}\right]\).

Find the rank of the matrix \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\\ 3 & 4 & 5\end{array}\right]\).

Find the rank of the matrix \(A=\left[\begin{array}{llll}1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1\end{array}\right]\).

Find the rank of the matrix \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 6 & 9 \\\ 1 & 2 & 3\end{array}\right]\).

Solve \(x+y+z=3\) \(2 x-y+z=2\) \(x-2 y+3 z=2\)

Solve \(x-y+z=4\) \(2 x+y-3 z=0\) \(x+y+z=2\)

Solve \(5 x+3 y+z=16\) \(2 x+y+3 z=19\) \(x+2 y+4 z=25\)

Solve \(2 x+8 y+5 z=5\) \(x+y+z=-2\) \(x+2 y-z=2\)

Solve \(x+2 y-z=6\) \(3 x-y-2 z=3\) \(4 x+3 y+z=9\)

Solve \(x+y+z=6\) \(x+2 y+3 z=10\) \(x+2 y+4 z=1\)

Solve \(x+y+z=4\) \(2 x+5 y-2 z=3\) \(x+7 y-7 z=5\)

Solve \(x+y+z=6\) \(x+2 y+3 z=14\) \(x+4 y+7 z=30\)

Solve \(5 x+3 y+7 z=4\) \(3 x+26 y+2 z=9\) \(7 x+2 y+10 z=5\)

Solve \(x-3 y-8 z=-10\) \(3 x+y-4 z=0\) \(2 x+5 y+6 z=13\)

Solve \(3 x-y+z=0\) \(-15 x+6 y-5 z=0\) \(5 x-2 y+2 z=0\)

Solve \(x+3 y-2 z=0\) \(2 x-y+4 z=0\) \(x-11 y+14 z=0\)

Solve \(3 x+2 y+7 z=0\) \(4 x-3 y-2 z=0\) \(5 x+9 y+23 z=0\)

Using matrix method, find the values of \(\lambda\) and \(\mu\) so that the system of equation \(2 x-3 y+5 z=12\) \(3 x+y+\lambda z=\mu\) \(x-7 y+8 z=17\) has i. Unique solution \\{Ans. \(\lambda \neq 2\\}\) ii. Infinite solutions \\{Ans. \(\lambda=2, \mu=7\\}\) iii. No solution \\{Ans. \(\lambda=2, \mu \neq 7\\}\)

If every element of a third order determinant of value \(\Delta\) is multiplied by 5, then find the value of new determinant.

If \(M\) is a \(3 \times 3\) matrix, where \(M^{T} M=I\) and \(\operatorname{det}(M)=1\), then prove that \(\operatorname{det}(M-I)=0\).

If \(A\) and \(B\) are two non-zero square matrices of the same order and \(A B=O\), then show that both \(A\) and \(B\) must be singular.