91Ó°ÊÓ

If \(A\) is unimodular, then which of the following is unimodular? a. \(-A\) b. \(A^{-1}\) c. \(\operatorname{adj} A\) d. \(\omega A\), where \(\omega\) is cube root of unity

If \(X=\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]\), then prove that \((p I+q X)^{m}=p^{\prime \prime \prime} I+m p^{m-1} q X, \forall\) \(p, q \in R\), where \(I\) is a two-rowed unit matrix and \(m \in N\).

The inverse of a skew-symmetric matrix of odd order is a. a symmetric matrix b. a skew symmetric c. diagonal matrix d. does not exist

If \(A=\left[\begin{array}{cc}1 & -1 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{cc}a & 1 \\ b & -1\end{array}\right]\) and \((A+B)^{2}=A^{2}+B^{2}+2 A B\), then a. \(a=-1\) b. \(a=1\) c. \(b=2\) d. \(b=-2\)

Let \(A\) and \(B\) be two \(2 \times 2\) matrices. Consider the statements (i) \(A B=O \Rightarrow A=\mathrm{O}\) or \(B=\mathrm{O}\) (ii) \(A B=I_{2} \Rightarrow A=B^{-1}\) (iii) \((A+B)^{2}=A^{2}+2 A B+B^{2}\) Then a. (i) and (ii) are false, (iii) is true b. (ii) and (iii) are false, (i) is true c. (i) is false, (ii) and (iii) are true d. (i) and (iii) are false, (ii) is true

If \(B, C\) are square matrices of order \(n\) and if \(A=B+C, B C\) \(=C B, C^{2}=O\), then without using mathematical induction, show that for any positive integer \(p, A^{p+1}=B^{p}[B+(p+1) C]\)

If \(A B=A\) and \(B A=B\), then which of the following is/are true? a. \(A\) is idempotent b. \(B\) is idempotent c. \(A^{T}\) is idempotent d. none of these

The equation \([1 x y]\left[\begin{array}{ccc}1 & 3 & 1 \\ 0 & 2 & -1 \\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{l}1 \\ x \\\ y\end{array}\right]=[0]\) has (i) for \(y=0\) (p) rational roots (ii) for \(y=-1\) (q) irrational roots (r) integral roots Then (i) (ii) a. (p) (r) b. (q) (p) c. (p) (q) d. (r) (p)

If \(D=\operatorname{diag}\left[d_{1}, d_{2}, \ldots, d_{n}\right]\), then prove that \(f(D)=\operatorname{diag}[f(d)\), \(\left.f\left(d_{2}\right), \ldots, f\left(d_{n}\right)\right]\), where \(f(x)\) is a polynomial with scalar coefficient.

If \(A=\frac{1}{3}\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{array}\right]\) is an orthogonal matrix, then a. \(a=-2\) b. \(a=2, b=1\) c. \(b=-1\) d. \(b=1\)