Chapter 8

If \(A\) and \(B\) are symmetric and commute, then which of the following is/are symmetric? a. \(A^{-1} B\) b. \(A B^{-1}\) c. \(A^{-1} B^{-1}\) d. None of these

If \(A\) is an orthogonal matrix, then \(A^{-1}\) equals a. \(A^{T}\) b. \(A\) c. \(A^{2}\) d. none of these

A skew-symmetric matrix \(A\) satisfies the relation \(A^{2}+I=O\), where \(I\) is a unit matrix then \(A\) is a. idempotent b. orthogonal c. of even order d. odd order

If \(Z\) is an idempotent matrix, th?n \((I+Z)^{n}\) a. \(I+2^{n} Z\) b. \(I+\left(2^{n}-1\right) Z\) c. \(I-\left(2^{\prime \prime}-1\right) Z\) d. none of these

If \(A B=A\) and \(B A=B\), then a. \(A^{2} B=A^{2}\) b. \(B^{2} A=B^{2}\) c. \(A B A=A\) d. \(B A B=B\)

If \(A\) and \(B\) are two matrices such that \(A B=B\) and \(B A=A\), then a. \(\left(A^{5}-B^{5}\right)^{3}=A-B\) b. \(\left(A^{5}-B^{5}\right)^{3}=A^{3}-B^{3}\) C. \(A-B\) is idempotent d. \(A-B\) is nilpotent

Let \(A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]\). Then a. \(A^{2}-4 A-5 I_{3}=O\) b. \(A^{-5}=\frac{1}{5}\left(A-4 I_{3}\right)\) c. \(A^{3}\) is not invertible d. \(A^{2}\) is invertible

If \(A\) is a nilpotent matrix of index 2 , then for any positive integer \(n, A(I+A)^{n}\) is equal to a. \(A^{-1}\) b. \(A\) c. \(A^{n}\) d. \(I_{n}\)

Let \(A\) be an \(n^{\text {th-order square matrix and } B \text { be its adjoint, then }}\) \(\mid A B+K I_{t}\) is (where \(K\) is a scalar quantity) a. \((|A|+K)^{n-2}\) b. \((|A|+K)^{n}\) C. \((|A|+K)^{\mu-1}\) d. none of these

If \(A^{2}=I\), then the value of \(\operatorname{det}(A-I)\) is (where \(A\) has order 3 ) a. 1 b. \(-1\) c. 0 d. cannot say anything

If \(A=\left[\begin{array}{lll}a & b & c \\ x & y & z \\ p & q & r\end{array}\right], B=\left[\begin{array}{rrr}q & -b & y \\ -p & a & -x \\\ r & -c & z\end{array}\right]\) and if \(A\) is invertible, then which of the following is not true? a. \(|A|=|B|\) b. \(|A|=-|B|\) c. \(|\operatorname{adj} A|=|\operatorname{adj} B|\) d. \(A\) is invertible if and only if \(B\) is invertible

If \(A\) and \(B\) are two non-singular matrices such that \(A B=C\), then \(|B|\) is equal to a. \(\frac{|C|}{|A|}\) b. \(\frac{|A|}{|C|}\) c. \(\mid C\) d. none of these

If \(A\) and \(B\) are squares matrices such that \(A^{2006}=O\) and \(A B=A\) \(+B\), then \(\operatorname{det}(B)\) equals a. 0 b. 1 c. \(-1\) d. none of these

If \(A\) is a non-diagonal involutory matrix, then a. \(A-I=O\) b. \(A+I=O\) c. \(A-I\) is non-zero singular \(\quad\) d. none of these

If \(A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1\end{array}\right]\) and \(A^{-1}=\left[\begin{array}{ccc}1 / 2 & 1 / 2 & 1 / 2 \\\ -4 & 3 & c \\ 5 / 2 & -3 / 2 & 1 / 2\end{array}\right]\), then the values of \(a\) and \(c\) are equal to a. 1,1 b. \(1,-1\) c. 1,2 d. \(-1,1\)

If \(A\) and \(B\) are two non-singular matrices of the same order such that \(B^{r}=I\), for some positive integer \(r>1\). Then \(A^{-1} B^{r-1} A-A^{-1} B^{-1} A=\) a. \(I\) b. \(2 l\) c. \(O\) d. \(-I\)

For two unimodular complex numbers \(z_{1}\) and \(z_{2}\), \(\left[\begin{array}{rr}\bar{z}_{1} & -z_{2} \\ \bar{z}_{2} & z_{1}\end{array}\right]^{-1}\left[\begin{array}{cc}z_{1} & z_{2} \\\ -\bar{z}_{2} & \bar{z}_{1}\end{array}\right]^{-1}\) is equal to a. \(\left[\begin{array}{cc}z_{1} & z_{2} \\ \bar{z}_{1} & \bar{z}_{2}\end{array}\right]\) b. \(\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) c. \(\left[\begin{array}{cc}1 / 2 & 0 \\ 0 & 1 / 2\end{array}\right]\) d. none of these

If \(A(\alpha, \beta)=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & e^{\beta}\end{array}\right]\), then \(A(\alpha, \beta)^{-1}\) is equal to a. \(A(-\alpha,-\beta)\) b. \(A(-\alpha, \beta)\) c. \(A(\alpha,-\beta)\) d. \(A(\alpha, \beta)\)

If \(A=\left[\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) and \(a^{2}+b^{2}+c^{2}+d^{2}=1\), then \(A^{-1}\) is equal to a. \(\left[\begin{array}{cc}a+i b & -c-i d \\ -c+i d & a-i b\end{array}\right]\) b. \(\left[\begin{array}{cc}a+i b & -c+i d \\ -c+i d & a-i b\end{array}\right]\) \(\mathbf{c} \cdot\left[\begin{array}{cc}a-i b & -c-i d \\ -c-i d & a+i b\end{array}\right]\) d. none of these

If \(A^{3}=O\), then \(I+A+A^{2}\) equals a. \(I-A\) b. \(\left(I+A^{\prime}\right)^{-1}\) c. \((I-A)^{-1}\) d. none of these

If \(A\) is order 3 square matrix such that \(|A|=2\), then \(|\operatorname{adj}(a d j(a d j A))|\) is a. 512 b. 256 c. 64 d. none of these

\((-A)^{-1}\) is always equal to (where \(A\) is \(n^{\text {tr-order square matrix })}\) a. \((-1)^{n} A^{-1}\) b. \(-A^{-1}\) c. \((-1)^{n-1} A^{-1}\) d. none of these

For each real \(x,-1

If \(\left[\begin{array}{cc}1 / 25 & 0 \\ x & 1 / 25\end{array}\right]=\left[\begin{array}{cc}5 & 0 \\ -a & 5\end{array}\right]^{-2}\), then the value of \(x\) is a. \(a / 125\) b. \(2 a / 125\) c. \(2 a / 25\) d. none of these

If \(A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\) and \(f(x)=\frac{1+x}{1-x}\), then \(f(A)\) is a. \(\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\) b. \(\left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]\) c. \(\left[\begin{array}{ll}-1 & -1 \\ -1 & -1\end{array}\right]\) d. none of these

If \(\boldsymbol{A}\) is a square matrix of order \(n\) such that \(|\operatorname{adj}(\operatorname{adj} A)|=|A|^{9}\), then the value of \(n\) can be \(\begin{array}{ll}\text { a. } 4 & \text { b. } 2\end{array}\) \(\begin{array}{ll}\text { c. either } 4 \text { or } 2 & \text { d. none of these }\end{array}\)

If \(\boldsymbol{A}=\left[\begin{array}{cc}1 & \tan x \\ -\tan x & 1\end{array}\right]\), then \(A^{T} A^{-1}\) is a. \(\left[\begin{array}{cc}-\cos 2 x & \sin 2 x \\ -\sin 2 x & \cos 2 x\end{array}\right]\) b. \(\left[\begin{array}{cc}\cos 2 x & -\sin 2 x \\ \sin 2 x & \cos 2 x\end{array}\right]\) \(\left[\begin{array}{cc}\cos 2 x & \cos 2 x \\ \cos 2 x & \sin 2 x\end{array}\right]\) d. none of these

If \(A=\left[\begin{array}{cc}0 & -\tan \alpha / 2 \\ \tan \alpha / 2 & 0\end{array}\right]\) and \(I\) is a \(2 \times 2\) unit matrix, then \((I-A)\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \sin \alpha\end{array}\right]\) is \(\begin{array}{ll}\text { a. }-I+A & \text { b. } I-A\end{array}\) c. \(-I-A\) d. none of these

The matrix \(X\) for which \(\left[\begin{array}{cc}1 & -4 \\ 3 & -2\end{array}\right] X=\left[\begin{array}{cc}-16 & -6 \\ 7 & 2\end{array}\right]\) is \(a \cdot\left[\begin{array}{ll}-2 & 4 \\ -3 & 1\end{array}\right]\) b. \(\left[\begin{array}{rr}-\frac{1}{5} & \frac{2}{5} \\ -\frac{3}{10} & \frac{1}{5}\end{array}\right]\) c. \(\left[\begin{array}{cc}-16 & -6 \\ 7 & 2\end{array}\right]\) d. \(\left[\begin{array}{cc}6 & 2 \\ \frac{11}{2} & 2\end{array}\right]\)

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

If \(A\) is singular matrix, then \(\operatorname{adj} A\) is a. singular b. non-singular c. symmetric d. not defined

The inverse of a diagonal matrix is a. a diagonal matrix b. a skew-symmetric matrix c. a symmetric matrix d. none of these

If \(P\) is non-singular matrix, then value of \(\operatorname{adj}\left(P^{-1}\right)\) in terms of \(P\) is a. \(P A P \mid\) b. \(P|P|\) c. \(P\) d. none of these

If adj \(B=A,|P|=|Q|=1\), then \(\operatorname{adj}\left(Q^{-1} B P^{-1}\right)\) is a. \(P Q\) b. \(Q A P\) c. \(P A Q\) d. \(P A^{-1} Q\)

If \(A\) is non-singular and \((A-2 I)(A-4 I)=O\), then \(\frac{1}{6} A+\frac{4}{3} A^{-1}\) is equal to a. \(O\) b. \(I\) c. \(2 I\) d. \(6 I\)

If \(A(a, \beta)=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & e^{\beta}\end{array}\right]\), then \(A(\alpha, \beta)^{-1}\) in terms of function of \(A\) is a. \(A(\alpha,-\beta) \quad\) b. \(A(-\alpha,-\beta)\) c. \(A(-\alpha, \beta)\) d. none of these

If \(A\) and \(B\) are two square matrices such that \(B=-A^{-1} B A\), then \((A+B)^{2}\) is equal to a. \(A^{2}+B^{2}\) b. \(O\) \(\begin{array}{ll}\text { c. } A^{2}+2 A B+B^{2} & \text { d. } A+B\end{array}\)

Let \(a\) and \(b\) be two real numbers such that \(a>1, b>1\). If \(A=\left(\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right)\), then \(\lim _{n \rightarrow \infty} A^{-n}\) is a. unit matrix b. null matrix c. \(2 I\) d. none of these

Let \(f(x)=\frac{1+x}{1-x}\). If \(A\) is matrix for which \(A^{3}=O\), then \(f(A)\) is \(\begin{array}{ll}\text { a. } I+A+A^{2} & \text { b. } I+2 A+2 A^{2}\end{array}\) \(\begin{array}{ll}\text { c. } I-A-A^{2} & \text { d. none of these }\end{array}\)

If \(A\) and \(B\) are two non-zero square matrices of the same order such that the product \(A B=O\), then a. both \(A\) and \(B\) must be singular b. exactly one of them must be singular c. both of them are non-singular d. none of these

If \(\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] A\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\), then \(A\) a. \(\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]\) b. \(\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\) c. \(\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\) d. \(-\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]\)

If \(A^{2}-A+I=0\), then the inverse of \(A\) is a. \(A^{-2}\) b. \(A+I\) c. \(I-A\) d. \(A-I\)

The number of solutions of the matrix equation \(X^{2}=\left[\begin{array}{ll}1 & 1 \\ 2 & 3\end{array}\right]\) is a. more than 2 b. 2 c. 0 d. 1

If \(A\) and \(B\) are symmetric matrices of the same order and \(X=A B+B A\) and \(Y=A B-B A\), then \((X Y)^{r}\) is equal to a. \(X Y\) b. \(Y X\) c. \(-Y X\) d. none of these

If \(A\) is a \(3 \times 3\) skew-symmetric matrix, then trace of \(A\) is equal to a. \(-1\) b. 1 c. \(|A|\) d. none of these

Elements of a matrix \(A\) of order \(10 \times 10\) are defined as \(a_{i j}=w^{i+j}\) (where \(w\) is cube root of unity), then trace \((A)\) of the matrix is a. 0 b. 1 c. 3 d. none of these

Let \(F(\alpha)=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]\), where \(\alpha \in R\). Then \((F(a))^{-1}\) is equal to a. \(F\left(a^{-1}\right)\) b. \(F(-\alpha)\) c. \(F(2 \alpha)\) d. none of these

If \(F(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\\ 0 & 0 & 1\end{array}\right]\) and \(G(y)=\left[\begin{array}{ccc}\cos y & 0 & \sin y \\ 0 & 1 & 0 \\ -\sin y & 0 & \cos y\end{array}\right]\) then \([F(x) G(y)]^{-1}\) is equal to a. \(F(-x) G(-y)\) b. \(G(-y) F(-x)\) c. \(F\left(x^{-1}\right) G\left(y^{-1}\right)\) d. \(G\left(y^{-1}\right) F\left(x^{-1}\right)\)

If \(A\) is a skew-symmetric matrix and \(n\) is odd positive integer, then \(A^{n}\) is a. a skew-symmetric matrix b. a symmetric matrix c. a diagonal matrix d. none of these

If \(A, B, A+I, A+B\) are idempotent matrices, then \(A B\) is equal to a. \(B A\) b. \(-B A\) c. \(I\) d. \(O\)