Chapter 8

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 \(A\) 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

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

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 \(p, A^{p+1}=B^{p}[B+(p+1) C]\)

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 \(A B=A\) 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

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.

The number of diagonal matrix \(A\) of order \(n\) for which \(A^{3}=A\) is a. I b. 0 c. \(2^{n}\) d. 3 "

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. \(b=1\)

If \(A=\left[\begin{array}{ll}a & b \\ 0 & a\end{array}\right]\) is \(n^{\text {th }}\) root of \(I_{2}\), then choose the correct statements: (i) if \(n\) is odd, \(a=1, b=0\) (ii) if \(n\) is odd, \(a=-1, b=0\) (iii) if \(n\) is even, \(a=1, b=0\) (iv) if \(n\) is even, \(a=-1, b=0\) a. \(\mathrm{i}, \mathrm{ii}, \mathrm{iii}\) b. ii, iii, iv c. i, ii, iii, iv d. i, iii, iv

Let \(A\) and \(B\) are two non-singular square matrices, \(A^{T}\) and \(B^{T}\) are the transpose matrices of \(A\) and \(B\), respectively, then which of the following are correct? a. \(B^{T} A B\) is symmetric matrix if \(A\) is symmetric b. \(B^{T} A B\) is symmetric matrix if \(B\) is symmetric c. \(B^{T} A B\) is skew-symmetric matrix for every matrix \(A\) d. \(B^{\tau} A B\) is skew-symmetric matrix if \(A\) is skew-symmetric

If \(A=\left[\begin{array}{cc}-1 & 1 \\ 0 & -2\end{array}\right]\), then prove that \(A^{2}+3 A+2 I=O\), hence find \(B\) and \(C\) matrices of order 2 with integer elements, if \(A=B^{3}+C^{3}\)

\(A\) is a \(2 \times 2\) matrix such that \(A\left[\begin{array}{c}1 \\\ -1\end{array}\right]=\left[\begin{array}{c}-1 \\ 2\end{array}\right]\) and \(A^{2}\left[\begin{array}{c}1 \\ -1\end{array}\right]=\left[\begin{array}{l}1 \\\ 0\end{array}\right]\). The sum of the elements of \(A\) is a. \(-1\) b. 0 c. 2 d. 5

If \(A(\theta)=\left[\begin{array}{cc}\sin \theta & i \cos \theta \\ i \cos \theta & \sin \theta\end{array}\right]\), then which of the following is not true? a. \(A(\theta)^{-1}=A(\pi-\theta)\) b. \(A(\theta)+A(\pi+\theta)\) is a null matrix c. \(A(\theta)\) is invertible for all \(\theta \in R\) d. \(A(\theta)^{-1}=A(-\theta)\)

Find the possible square roots of the two-rowed unit matrix \(I .\)

The product of matrices \(A=\left[\begin{array}{cc}\cos ^{2} \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin ^{2} \theta\end{array}\right]\) and \(B=\left[\begin{array}{cc}\cos ^{2} \phi & \cos \phi \sin \phi \\ \cos \phi \sin \phi & \sin ^{2} \phi\end{array}\right]\) is a null matrix if \(\theta-\phi=\) a. \(2 n \pi, n \in Z\) b. \(n \frac{\pi}{2}, n \in Z\) c. \((2 n+1) \frac{\pi}{2}, n \in Z\) d. \(n \pi, n \in Z\)

If \(A\) is a matrix such that \(A^{2}+A+2 I=O\), then which of the following is/are true? a. \(A\) is non-singular b. \(A\) is symmetric c. A cannot be skew-symmetric d. \(A^{-1}=-\frac{1}{2}(A+I)\)

If \(A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]\), then show that \(A^{2}-4 A-5 I=O\), where \(I\) and 0 are the unit matrix and the null matrix of order 3, respectively. Use this result to find \(A^{-1}\).

If \(A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]\), then a. \(\operatorname{adj}(\operatorname{adj} A)=A\) b. \(|\operatorname{adj}(\operatorname{adj} A)|=1\) c. \(|\operatorname{adj} A|=1\) d. none of these

If \(S\) is a real skew-symmetric matrix, then prove that \(I-S\) is non-singular and the matrix \(A=(I+S)(I-S)^{-1}\) is orthogonal.

If \(A=\left[a_{i j}\right]_{4 \times 4}\), such that \(a_{i j}=\left\\{\begin{array}{ll}2, & \text { when } i=j \\ 0, & \text { when } i \neq j\end{array}\right.\), then \(\left\\{\frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj} A))}{7}\right\\}\) is (where \(\\{\cdot\\}\) represents fractional part function) a. \(1 / 7\) b. \(2 / 7\) c. \(3 / 7\) d. none of these

If \(\left(\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right)\left(\begin{array}{cc}1 & \tan \theta \\ -\tan \theta & 1\end{array}\right)^{-1}=\left[\begin{array}{cc}a & -b \\ b & a\end{array}\right]\), then a. \(a=\cos 2 \theta\) b. \(a=1\) c. \(b=\sin 2 \theta\) d. \(b=-1\)

If \(B\) and \(C\) are non-singular matrices and \(O\) is null matrix, then show that \(\left[\begin{array}{cc}A & B \\ C & O\end{array}\right]^{-1}=\left[\begin{array}{cc}O & C^{-1} \\ B^{-1} & -B^{-1} A C^{-1}\end{array}\right]\)

If \(\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha\end{array}\right]\) is to be the square root of two-rowed unit matrix, then \(\alpha, \beta\) and \(\gamma\) should satisfy the relation a. \(1-\alpha^{2}+\beta \gamma=0\) b. \(\alpha^{2}+\beta \gamma-1=0\) c. \(1+\alpha^{2}+\beta \gamma=0\) d. \(1-a^{2}-\beta \gamma=0\)

If \(A^{-1}=\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 3 & 1 \\ 0 & 0 & -1 / 3\end{array}\right]\), then a. \(|A|=-1\) b. \(\operatorname{adj} A=\left[\begin{array}{ccc}-1 & 1 & -2 \\ 0 & -3 & -1 \\\ 0 & 0 & 1 / 3\end{array}\right]\) c. \(A=\left[\begin{array}{ccc}1 & 1 / 3 & 7 \\ 0 & 1 / 3 & 1 \\ 0 & 0 & -3\end{array}\right]\) d. \(A=\left[\begin{array}{ccc}1 & -1 / 3 & -7 \\ 0 & -3 & 0 \\ 0 & 0 & 1\end{array}\right]\)

Show that every square matrix \(A\) can be uniquely expressed as \(P+i Q\), where \(P\) and \(Q\) are Hermitian matrices.

If \(A\) is a square matrix such that \(A^{2}=A\), then \((I+A)^{3}-7 A\) is a. \(3 I\) b. \(O\) c. \(I\) d. \(2 I\)

If \(B\) is an idempotent matrix, and \(A=I-B\), then a. \(A^{2}=A\) b. \(A^{2}=I\) c. \(A B=O\) d. \(B A=O\)

Express \(A\) as the sum of a Hermitian and a skew-Hermitian matrix, where \(A=\left[\begin{array}{ccc}2+3 i & 2 & 5 \\ -3-i & 7 & 3-i \\ 3-2 i & i & 2+i\end{array}\right]\).

If \(A\) and \(B\) are square matrices of order \(n\), then \(A-\lambda I\) and \(B-\lambda I\) commute for every scalar \(\lambda\), only if a. \(A B=B A\) b. \(A B+B A=O\) c. \(A=-B\) d. none of these

Which of the following statements is/are true about square matrix \(A\) of order \(n ?\) a. \((-A)^{-1}\) is equal to \(-A^{-1}\) when \(n\) is odd only. b. If \(A^{n}=O\), then \(I+A+A^{2}+\cdots+A^{n-1}=(I-A)^{-1}\). c. If \(A\) is skew-symmetric matrix of odd order, then its inverse does not exist. d. \(\left(A^{7}\right)^{-1}=\left(A^{-1}\right)^{T}\) holds always.

Matrix \(A\) such that \(A^{2}=2 A-I\), where \(I\) is the identity matrix, then for \(n \geq 2, A^{n}\) is equal to a. \(2^{n-1} A-(n-1) I\) b. \(2^{n-1} A-I\) c. \(n A-(n-1) I\) d. \(n A-I\)

Let \(A=\left[\begin{array}{ll}0 & \alpha \\ 0 & 0\end{array}\right]\) and \((A+1)^{50}-50 A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\). Then the value of \(a+b+c+d\) is a. 2 b. c. 4 d. none of these

If \(A=\left(a_{i j}\right)_{n \times n}\) and \(f\) is a function, we define \(f(A)=\left(f\left(a_{i j}\right)\right)_{n \times n^{\circ}}\) Let \(A=\left(\begin{array}{cc}\pi / 2-\theta & \theta \\ -\theta & \pi / 2-\theta\end{array}\right) .\) Then a. \(\sin A\) is invertible b. \(\sin A=\cos A\) c. \(\sin A\) is orthogonal d. \(\sin (2 A)=2 \sin A \cos A\)

If \(A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right]\) and \(B=\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]\), then \(A^{8}\) equals a. \(4 B\) b. \(128 B\) c. \(-128 B\) d. \(-64 B\)

If \(\left[\begin{array}{rr}2 & -1 \\ 1 & 0 \\ -3 & 4\end{array}\right] A=\left[\begin{array}{rrr}-1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15\end{array}\right]\), then sum of all the elements of matrix \(A\) is a. 0 b. 1 c. 2 d. \(-3\)

If \(A\) and \(B\) are two invertible matrices of the same order, then \(\operatorname{adj}(A B)\) is equal to a. \(\operatorname{adj}(B) \operatorname{adj}(A)\) b. \(|B||A| B^{-1} A^{-1}\). c. \(|B||A| A^{-1} B^{-1}\) d. \(|A||B|(A B)^{-1}\)

If \(A=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 & 1+i \\ 1-i & 1\end{array}\right]\), then \(A\left(\bar{A}^{T}\right)\) equals a. \(O\) b. \(I\) c. \(-I\) d. \(2 I\)

Identify the incorrect statement in respect of two square matrices \(A\) and \(B\) conformable for sum and product: a. \(t_{c}(A+B)=t_{c}(A)+t_{r}(B)\) b. \(t_{r}(\alpha A)=a t_{r}(A), \alpha \in R\) c. \(t_{r}\left(A^{T}\right)=t_{r}(A)\) d. none of these.

\(A\) is an involuntary matrix given by \(A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4\end{array}\right]\), then the inverse of \(A / 2\) will be a. \(2 A\) b. \(\frac{A^{-1}}{2}\) c. \(\frac{A}{2}\) d. \(A^{2}\)

If \(a, \beta, \gamma\) are three real numbers and \(A=\left[\begin{array}{ccc}1 & \cos (\alpha-\beta) & \cos (\alpha-\gamma) \\ \cos (\beta-\alpha) & 1 & \cos (\beta-\gamma) \\ \cos (\gamma-\alpha) & \cos (\gamma-\beta) & 1\end{array}\right]\), then which of following is/are true? a. \(A\) is singular b. \(A\) is symmetric c. \(A\) is orthogonal d. \(A\) is not invertible

If \(A\) is a non-singular matrix such that \(A A^{T}=A^{T} A\) and \(B=A^{-1} A^{\tau}\), then matrix \(B\) is a. involuntary b. orthogonal c. idempotent d. none of these

Let \(A=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\). Then which of following is not true? a. \(\lim _{n \rightarrow \infty} \frac{1}{n^{2}} A^{-n}=\left[\begin{array}{cc}0 & 0 \\ -1 & 0\end{array}\right]\) b. \(\lim _{n \rightarrow \infty} \frac{1}{n} A^{-n}=\left[\begin{array}{cc}0 & 0 \\ -1 & 0\end{array}\right]\) c. \(A^{-w}=\left[\begin{array}{cc}1 & 0 \\ -n & 1\end{array}\right] \forall n \neq N\) d. none of these

If \(P\) is an orthogonal matrix and \(Q=P A P^{T}\) and \(x=P^{T} Q^{1000} P\), then \(x^{-1}\) is, where \(A\) is involutary matrix a. \(A\) b. \(I\) c. \(A^{1000}\) d. none of these

If \(C\) is skew-symmetric matrix of order \(n\) and \(X\) is \(n \times 1\) column matrix, then \(X^{T} C X\) is a. singular b. non-singular c. invertible d. non-invertible

If \(n^{i \text { - }}\)-order square matrix \(A\) is a orthogonal, then, \(|\operatorname{adj}(\operatorname{adj} A)|\) is a. always \(-1\) if \(n\) is even b. always 1 if \(n\) is odd c. always 1 d. none of these

If \(S=\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right]\) and \(A=\left[\begin{array}{ccc}b+c & c+a & b-c \\ c-b & c+b & a-b \\ b-c & a-c & a+b\end{array}\right](a, b, c \neq 0)\), then \(S A S^{-1}\) is a. symmetric matrix b. diagonal matrix c. invertible matrix d. singular matrix

If \(D_{1}\) and \(D_{2}\) are two \(3 \times 3\) diagonal matrices, then which of the following is/are true? a. \(D_{1} D_{2}\) is diagonal matrix b. \(D_{1} D_{2}=D_{2} D_{1}\) c. \(D_{1}^{2}+D_{2}^{2}\) is a diagonal matrix d. none of these