Chapter 5

If \(A\) and \(B\) are symmetric matrices and \(A B=B A\), then \(A^{-1} B\) is a (A) symmetric matrix (B) skew-symmetric matrix (C) identity matrix (D) None of these

If the product of the matrix \(B=\left[\begin{array}{ccc}2 & 6 & 4 \\ 1 & 0 & 1 \\ -1 & 1 & -1\end{array}\right]\) with a matrix \(A\) has inverse \(C=\left[\begin{array}{ccc}-1 & 0 & 1 \\ 1 & 1 & 3 \\\ 2 & 0 & 2\end{array}\right]\), then \(A^{-1}\) equals (A) \(\left[\begin{array}{ccc}-3 & -5 & 5 \\ 0 & 9 & 14 \\ 2 & 2 & 6\end{array}\right]\) (B) \(\left[\begin{array}{ccc}-3 & 5 & 5 \\ 0 & 0 & 9 \\ 2 & 14 & 16\end{array}\right]\) (C) \(\left[\begin{array}{ccc}-3 & -5 & -5 \\ 0 & 9 & 2 \\ 2 & 14 & 6\end{array}\right]\) (D) \(\left[\begin{array}{ccc}-3 & -3 & -5 \\ 0 & 9 & 2 \\ 2 & 14 & 6\end{array}\right]\)

If \(A=\left[\begin{array}{cc}\alpha & 2 \\ 2 & \alpha\end{array}\right]\) and \(\left|A^{3}\right|=125\) then the value of \(\alpha\) is (A) \(\pm 1\) (B) \(\pm 2\) (C) \(\pm 3\) (D) \(\pm 5\)

If \(A\) is an involutory matrix and \(I\) is unit matrix of the same order then, \((I-A)(I+A)=\) (A) 0 (B) \(A\) (C) \(I\) (D) \(2 A\)

Matrix \(A\) is such that \(A^{2}=2 A-I\), where \(I\) is unit matrix then for \(n \geq 2, A^{n}=\) (A) \(n A-(n-1) I\) (B) \(n A-I\) (C) \(2^{n-1} A-(n-1) I\) (D) \(2^{n-1} A-I\)

If \(A B=A\) and \(B A=B\), where \(A\) and \(B\) are square matrices, then (A) \(B^{2}=B\) and \(A^{2}=A\) (B) \(B^{2}=A\) and \(A^{2}=B\) (C) \(A B=B A\) (D) None of these

If \(A\) is a square matrix, \(B\) is a singular matrix of same order, then for a positive integer \(n,\left(A^{-1} B A\right)^{n}\) equals (A) \(A^{-\pi} B^{n} A^{n}\) (B) \(A^{n} B^{n} A^{-n}\) (C) \(A^{-1} B^{n} A\) (D) \(n\left(A^{-1} B A\right)\)

If \(A=\left(\begin{array}{rr}3 & -4 \\ 1 & -1\end{array}\right)\) then \(A^{n}\) equals (A) \(\left(\begin{array}{cc}3 n & -4 n \\ n & -n\end{array}\right)\) (B) \(\left(\begin{array}{cc}3^{n} & 4^{n}(-1)^{n} \\ 1^{n} & (-1)^{n}\end{array}\right)\) (C) \(\left(\begin{array}{cc}3+n & -(4+n) \\ n & -n\end{array}\right)\) (D) None of these

The matrix \(A=\left[\begin{array}{cc}1 / \sqrt{2} & 1 / \sqrt{2} \\ -1 / \sqrt{2} & -1 / \sqrt{2}\end{array}\right]\) is (A) unitary (B) orthogonal (C) nilpotent (D) involutary

If \(E(\theta)=\left[\begin{array}{cc}\cos ^{2} \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin ^{2} \theta\end{array}\right]\) and \(\theta\) and \(\phi\) differ by an odd multiple of \(\frac{\pi}{2}\), then \(E(\theta) E(\phi)\) is a (A) null matrix (B) unit matrix (C) diagonal matrix (D) None of these

If \(A=\left[\begin{array}{cc}1 & \tan \theta / 2 \\ -\tan \theta / 2 & 1\end{array}\right]\) and \(A B=I\), then \(B=\) (A) \(\cos ^{2} \frac{\theta}{2} A\) (B) \(\cos ^{2} \frac{\theta}{2} A^{T}\) (C) \(\cos ^{2} \frac{\theta}{2} I\) (D) None of these

For each real number \(x\) such that \(-1

The inverse of a skew symmetric matrix of odd order is (A) a symmetric matrix (B) a skew symmetric matrix (C) diagonal matrix (D) does not exist

The number of solutions of equations \(x_{2}-x_{3}=1,-x_{1}+\) \(2 x_{3}=2, x_{1}-2 x_{2}=3\) is (A) zero (B) one (C) two (D) infinite

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) \(1-\alpha^{2}-\beta \gamma=0\) (C) \(1-\alpha^{2}+\beta \gamma=0\) (D) \(\alpha^{2}+\beta \gamma-1=0\)

Let \(A\) and \(B\) be two symmetric matrices of order \(3 .\) Statement 1: \(A(B A)\) and \((A B) A\) are symmetric matrices. Statement \(2: A B\) is symmetric matrix if matrix multiplication of \(A\) with \(B\) is commutative. (A) Statement 1 is false, Statement 2 is true (B) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1 (C) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is true, Statement 2 if false

Let \(A=\left(\begin{array}{ccc}1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1\end{array}\right) .\) If \(u_{1}\) and \(u_{2}\) are column matrices such that \(A u_{1}=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)\) and \(A u_{2}=\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)\), then \(u_{1}+u_{2}\) is (A) \(\left(\begin{array}{r}-1 \\ 1 \\ 0\end{array}\right)\) (B) \(\left(\begin{array}{r}-1 \\ 1 \\ -1\end{array}\right)\) (C) \(\left(\begin{array}{r}-1 \\ -1 \\ 0\end{array}\right)\) (D) \(\left(\begin{array}{r}1 \\ -1 \\ -1\end{array}\right)\)

Let \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\), be a \(2 \times 2\) matrix where \(a, b, c, d\) take the values 0 or 1 only. The number of such matrices which have inverses is: (A) 8 (B) 7 (C) 6 (D) 5

Let \(A\) be a \(2 \times 2\) matrix with real entries. Let \(I\) be the \(2 \times 2\) identity matrix. Denote by \(\operatorname{tr}(A)\), the sum of diagonal entries of \(A\). Assume that \(A^{2}=I\) Statement 1: If \(A \neq I\) and \(A \neq-I\), then \(\operatorname{det} A=-1\). Statement 2: If \(A \neq I\) and \(A \neq-I\), then \(\operatorname{tr}(A) \neq 0\). (A) Statement 1 is false, Statement 2 is true (B) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1 (C) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is true, Statement 2 is false

Let \(A\) be a square matrix all of whose entries are integers. Then which one of the following is true? (A) If det \(A=\pm 1\), then \(A^{-1}\) exists but all its entries are not necessarily integers (B) If det \(A \neq \pm 1\), then \(A^{-1}\) exists and all its entries are non- integers (C) If det \(A=\pm 1\), then \(A^{-1}\) exists and all its entries are integers (D) If \(\operatorname{det} A=\pm 1\), then \(A^{-1}\) need not exist

If \(A=\left[\begin{array}{ccc}1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1\end{array}\right]\), then det. \((\operatorname{adj}(\operatorname{adj} A)\) ) is (A) \((14)^{4}\) (B) \((14)^{3}\) (C) \((14)^{2}\) (D) \((14)^{1}\)

If \(a b c=p\) and \(A=\left[\begin{array}{lll}a & b & c \\ c & a & b \\ b & c & a\end{array}\right]\) such that \(A A^{\prime}=I\), then \(a, b, c\) are the roots of the equation (A) \(x^{3}+p=0\) (B) \(x^{3} \pm x^{2}+p=0\) (C) \(x^{3} \pm 3 x^{2}+p=0\) (D) \(x^{3} \pm 2 x^{2}+p=0\)

If \(A\) is a singular matrix, then \(\operatorname{adj} A\) is (A) non-singular (B) singular (C) symmetric (D) not defined

Matrix \(A\) is such that \(A^{2}=2 A-I\), where \(I\) is unit matrix then for \(n \geq 2, A^{n}=\) (A) \(n A-(n-1) I\) (B) \(n A-I\) (C) \(2^{n-1} A-(n-1) I\) (D) \(2^{n-1} A-I\)

For each real number \(x\) such that \(-1

The inverse of a skew-symmetric matrix of odd order is (A) a symmetric matrix (B) a skew-symmetric matrix (C) diagonal matrix (D) does not exist

If \(A\) is an invertible matrix, then (A) \(\operatorname{adj} A^{\prime}=(\operatorname{adj} A)^{\prime}\) (B) \(\operatorname{adj} A^{\prime}=\operatorname{adj} A\) (C) adj \(A^{\prime}=A^{\prime}\) (D) None of these

If \(A\) is a non-singular square matrix of order \(n\), then adj \((\operatorname{adj} A)\) is equal to (A) \(|A|^{n} A\) (B) \(|A|^{n-1} A\) (C) \(|A|^{n-2} A\) (D) None of these

If \(D=\operatorname{diag}\left(a_{1} a_{2} a_{3} \ldots a_{n}\right)\), where \(a_{i} \neq 0\) for all \(i=1\), \(2, \ldots, n\), then \(D^{-1}\) is equal to (A) \(I_{n}\) (B) \(D\) (C) diag \(\left(\begin{array}{lll}a_{1}^{-1} & a_{2}^{-1} & a_{3}^{-1} & \ldots & a_{n}^{-1}\end{array}\right)\) (D) None of these

If \(A\) is a non-singular matrix such that \(A A^{\prime}=A^{\prime} A\) and \(B=A^{-1} A^{\prime}\), then \(B B^{\prime}\) is (A) \(I\) (B) \(B^{-1}\) (C) \(\left(B^{-1}\right)^{\prime}\) (D) None of these

If \(A^{3}=0\) and \(A^{n} \neq I\) for \(n=1,2\) then \((I-A)^{-1}\) is (A) \(I+A\) (B) \(I+A+A^{2}\) (C) \(I-A+A^{2}\) (D) None of these

\(A\) and \(B\) are two non-singular matrices of the same order such that \(A^{n}=I\) for some positive integer \(n>1\). Then, \(B A^{n-1} B^{-1}-B A^{-1} B^{-1}\) (A) is a null matrix (B) is an identity matrix (C) a singular matrix (D) None of these

A skew-symmetric matrix \(A\) satisfies the relation \(A^{2}+\) \(I=0\), where \(I\) is a unit matrix. Then, \(A\) is (A) Idempotent matrix (B) Orthogonal matrix (C) Nilpotent matrix (D) None of these

Let \(A\) be an \(n \times n\) matrix such that \(A^{n}=\alpha A\), where \(\alpha\) is a real number different from 1 and \(-1\). Then, the matrix \(A+I_{n}\) is (A) singular (B) non-singular, i.e., invertible (C) scalar matrix (D) None of these

If adj \(B=A\) and \(P, Q\) are two unimodular matrices, i.e., \(|P|=1=|Q|\), then \(\left(Q^{-1} B P^{-1}\right)^{-1}\) is equal to (A) \(P A Q\) (B) \(P B Q\) (C) \(Q A P\) (D) \(Q B P\)

If \(A=\left[\begin{array}{cc}1 & \frac{\alpha}{n} \\ -\frac{\alpha}{n} & 1\end{array}\right]\), then (A) \(\lim _{n \rightarrow \infty} A^{n}=0\) (B) \(\lim _{n \rightarrow \infty} \frac{1}{n} A^{n}=0\) (C) \(\lim _{n \rightarrow \infty} \frac{1}{n^{2}} A^{n}=0\) (D) None of these

If \(A^{k}=0\) for some value of \(k\) and \((I-A)^{p}=I+A+A^{2}+\) \(\ldots+A^{k-1}\), then \(p\) is (A) \(-1\) (B) \(-2\) (C) \(-3\) (D) None of these

If \(A\) satisfies the equation \(x^{3}-5 x^{2}+4 x+k I=0\), then \(A^{-1}\) exists if (A) \(k \neq-1\) (B) \(k \neq 0\) (C) \(k \neq 1\) (D) None of these

If \(M\) is a \(3 \times 3\) matrix, where \(M^{\prime} M=I\) and det \(M=1\), then det \((M-I)=\) (A) 0 (B) 1 (C) \(-1\) (D) None of these

Let \(A\) and \(B\) be two non-null square matrices. If the product \(A B\) is a null matrix, then (A) \(A\) is singular (B) \(B\) is singular (C) \(A\) is non-singular (D) \(B\) is non-singular

The rank of the matrix \(\left[\begin{array}{ccc}-1 & 2 & 5 \\ 2 & -4 & a-4 \\\ 1 & -2 & a+1\end{array}\right]\) is (A) 1 if \(a=6\) (B) 2 if \(a=1\) (C) 3 if \(a=2\) (D) 1 if \(a=-6\)

The system of equations \(2 x-3 y+6 z-5 t=3, y-4 z+\) \(t=1,4 x-5 y+8 z-9 t=k\) has (A) no solution if \(k \neq 7\) (B) no solution if \(k=7\) (C) infinite solutions if \(k \neq 7\) (D) infinite solutions if \(k=7\)

Which of the following is correct? (A) If \(A\) is a symmetric matrix, then \(A^{n}\) is symmetric, \(n \in N\) (B) If \(A\) is a skew-symmetric matrix then \(A^{n}\) is symmetric if \(n\) is even, \(\overline{n \in N}\) (C) If \(A\) is a skew-symmetric matrix then \(A^{n}\) is skew-symmetric if \(n\) is odd, \(n \in N\) (D) All of these

If \(A\) is a non-singular matrix, then (A) \(A^{-1}\) is symmetric if \(A\) is symmetric (B) \(A^{-1}\) is skew-symmetric if \(A\) is symmetric (C) \(\left|A^{-1}\right|=|A|\) (D) \(\left|A^{-1}\right|=|A|^{-1}\)

Which of the following is true? (A) Transpose of an orthogonal matrix is also orthogonal (B) Every orthogonal matrix is non-singular (C) Product of the two orthogonal matrices is also orthogonal (D) Inverse of an orthogonal matrix is also orthogonal

Suppose, \(a, b, c\) are real numbers such that \(a b c=1\). If the matrix \(A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]\) is such that \(A^{\prime} A=I\), then the value of \(a^{3}+b^{3}+c^{3}\) is (A) 1 (B) 2 (C) 3 (D) 4

Let \(A, B, C\) be \(2 \times 2\) matrices with entries from the set of real numbers. Define operation '*' as follows $$ A * B=\frac{1}{2}(A B+B A), \text { then } $$ (A) \(A * I=A\) (B) \(A * A=A^{2}\) (C) \(A * B=B * A\) (D) \(A *(B+C)=A * B+A * C\)

If \(A\) and \(B\) are two matrices such that \(A B=B A\), then \(\forall n \in N\) (A) \(A^{n} B=B A^{n}\) (B) \((A B)^{n}=A^{n} B^{n}\) (C) \((A+B)^{n}={ }^{n} C_{0} A^{n}+{ }^{n} C_{1} A^{n-1} B+{ }^{n} C_{2} A^{n-2} B^{2}+\ldots+\) \({ }^{n} C_{n} B^{n}\) (D) \(A^{2 n}-B^{2 n}=\left(A^{n}-B^{n}\right)\left(A^{n}+B^{n}\right)\)

If \(A^{-1}=\left[\begin{array}{rrr}1 & 0 & -2 \\ -2 & 1 & 0 \\ -1 & 1 & 0\end{array}\right]\), then (A) \(|A|=2\) (B) \(a d j \cdot A=\left[\begin{array}{ccc}\frac{1}{2} & 0 & -1 \\ -1 & \frac{1}{2} & 0 \\ -\frac{1}{2} & \frac{1}{2} & 0\end{array}\right]\) (C) \(|a d j, A|=4\) (D) \(\left|A^{\prime}\right|=\frac{1}{2}\)

If \(A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0\end{array}\right]\), then (A) \(A^{3}=I\) (B) \(A^{-1}=A^{2}\) (C) \(A^{n}=A, \forall n \neq 4\) (D) None of these