Chapter 8

The matrix \(A=\left[\begin{array}{ccc}-5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1\end{array}\right]\) is a. idempotent matrix b. involutory matrix c. nilpotent matrix d. none of these

If \(A\) is symmetric as well as skew-symmetric matrix, then \(A\) is a. diagonal matrix b. null matrix c. triangular matrix d. none of these

If \(A\) and \(B\) are square matrices of the same order and \(A\) is nonsingular, then for a positive integer \(n,\left(A^{-1} B A\right)^{n}\) is equal to a. \(A^{\cdots} B^{n} A^{n}\) b. \(A^{n} B^{n} A^{-n}\) c. \(A^{-1} B^{\prime \prime} A\) d. \(n\left(A^{-1} B A\right)\)

Which of the following is an orthogonal matrix? a. \(\left[\begin{array}{ccc}6 / 7 & 2 / 7 & -3 / 7 \\ 2 / 7 & 3 / 7 & 6 / 7 \\\ 3 / 7 & -6 / 7 & 2 / 7\end{array}\right]\) b. \(\left[\begin{array}{ccc}6 / 7 & 2 / 7 & 3 / 7 \\ 2 / 7 & -3 / 7 & 6 / 7 \\\ 3 / 7 & 6 / 7 & -2 / 7\end{array}\right]\) c. \(\left[\begin{array}{ccc}-6 / 7 & -2 / 7 & -3 / 7 \\ 2 / 7 & 3 / 7 & 6 / 7 \\\ -3 / 7 & 6 / 7 & 2 / 7\end{array}\right]\) d. \(\left[\begin{array}{ccc}6 / 7 & -2 / 7 & 3 / 7 \\ 2 / 7 & 2 / 7 & -3 / 7 \\\ -6 / 7 & 2 / 7 & 3 / 7\end{array}\right]\)

Given that matrix \(A=\left[\begin{array}{lll}x & 3 & 2 \\ 1 & y & 4 \\ 2 & 2 & z\end{array}\right]\). If \(x y z=60\) and \(8 x+4 y+3 z\) \(=20\), then \(A(\) adj \(A\) ) is equal to a. \(\left[\begin{array}{ccc}64 & 0 & 0 \\ 0 & 64 & 0 \\ 0 & 0 & 64\end{array}\right]\) b. \(\left[\begin{array}{ccc}88 & 0 & 0 \\ 0 & 88 & 0 \\ 0 & 0 & 88\end{array}\right]\) c. \(\left[\begin{array}{ccc}68 & 0 & 0 \\ 0 & 68 & 0 \\ 0 & 0 & 68\end{array}\right]\) d. \(\left[\begin{array}{ccc}34 & 0 & 0 \\ 0 & 34 & 0 \\ 0 & 0 & 34\end{array}\right]\)

Let \(A+2 B=\left[\begin{array}{ccc}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{array}\right]\) and \(2 A-B=\left[\begin{array}{ccc}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{array}\right]\). Then \(\operatorname{tr}(A)-\operatorname{tr}(B)\) has the value equal to a. 0 b. 1 c. 2 d. none

Let \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 0 & 5 \\ 0 & 2 & 1\end{array}\right]\) and \(B=\left[\begin{array}{c}0 \\ -3 \\\ 1\end{array}\right]\). Which of the following is true? a. \(A X=B\) has a unique solution b. \(A X=B\) has exactly three solutions c. \(A X=B\) has infinitely many solutions d. \(A X=B\) is inconsistent

Consider three matrices \(A=\left[\begin{array}{ll}2 & 1 \\ 4 & 1\end{array}\right], B=\left[\begin{array}{ll}3 & 4 \\ 2 & 3\end{array}\right]\) and \(C=\left[\begin{array}{cc}3 & -4 \\ -2 & 3\end{array}\right]\). Then the value of the sum \(\operatorname{tr}(A)+\operatorname{tr}\left(\frac{A B C}{2}\right)\) \(+\operatorname{tr}\left(\frac{A(B C)^{2}}{4}\right)+\operatorname{tr}\left(\frac{A(B C)^{3}}{8}\right)+\cdots+\infty\) is a. 6 b. 9 c. 12 d. none