Chapter 18

Let \(\alpha\) be a root of the equation \(x^{2}+x+1=0\) and the matrix \(A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & \alpha & \alpha^{2} \\ 1 & \alpha^{2} & \alpha^{4}\end{array}\right]\), then the matrix \(A^{31}\) is equal to: [Jan. 7, 2020 (I)] (a) \(A\) (b) \(I_{3}\) (c) \(A^{2}\) (d) \(A^{3}\)

If \(A=\left[\begin{array}{cc}\cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta\end{array}\right],\left(\theta=\frac{\pi}{24}\right)\) and \(A^{5}=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\), where \(i=\sqrt{-1}\), then which one of the following is not true? (a) \(0 \leq a^{2}+b^{2} \leq 1\) (b) \(a^{2}-d^{2}=0\) (c) \(a^{2}-c^{2}=1\) (d) \(a^{2}-b^{2}=\frac{1}{2}\)

Let \(A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right], x \in \mathbf{R}\) and \(A^{4}=\left[a_{i j}\right]\). If \(a_{11}=109\), then \(a_{22}\) is equal to [NA Sep. 03,2020 (I)]

Let \(\mathrm{A}=\left[\begin{array}{cc}\cos \alpha-\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right],(\alpha \in \mathrm{R})\) such that \(\mathrm{A}^{32}=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]\). Then a value of \(\alpha\) is : \(\quad\) [April 8, 2019 (I)] (a) \(\frac{\pi}{32}\) (b) 0 (c) \(\frac{\pi}{64}\) (d) \(\frac{\pi}{16}\)

Let \(P=\left[\begin{array}{lll}1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1\end{array}\right]\) and \(\mathrm{Q}=\left[\mathrm{q}_{\mathrm{ij}}\right]\) be two \(3 \times 3\) matrices such that \(\mathrm{Q}-\mathrm{P}^{5}=\mathrm{I}_{3}\). Then \(\frac{\mathrm{q}_{21}+\mathrm{q}_{31}}{\mathrm{q}_{32}}\) is equal to : [Jan. \(12,2019(\mathrm{I})]\) (a) 10 (b) 135 (c) 15 (d) 9

Let \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{array}\right]\) and \(B=A^{20}\). Then the sum of the elements of the first column of \(B\) is? [Online April 16, 2018] (a) 211 (b) 210 (c) 231 (d) 251

If \(A=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]\), then which one of the following statements is not correct? \(\quad\) [Online April 10, 2015] (a) \(\mathrm{A}^{2}+\mathrm{I}=\mathrm{A}\left(\mathrm{A}^{2}-\mathrm{I}\right)\) (b) \(\mathrm{A}^{4}-\mathrm{I}=\mathrm{A}^{2}+\mathrm{I}\) (c) \(\mathrm{A}^{3}+\mathrm{I}=\mathrm{A}\left(\mathrm{A}^{3}-\mathrm{I}\right)\) (d) \(\mathrm{A}^{3}-\mathrm{I}=\mathrm{A}(\mathrm{A}-\mathrm{I})\)

If \(A=\left[\begin{array}{ccc}1 & 2 & x \\ 3 & -1 & 2\end{array}\right]\) and \(B=\left[\begin{array}{l}y \\ x \\ 1\end{array}\right]\) be such that \(\mathrm{AB}=\left[\begin{array}{l}6 \\ 8\end{array}\right]\), then: \(\quad\) [Online April 12, 2014] (a) \(y=2 x\) (b) \(y=-2 x\) (c) \(y=x\) (d) \(y=-x\)

If \(p, q, r\) are 3 real numbers satisfying the matrix equation, \([p q r]\left[\begin{array}{lll}3 & 4 & 1 \\ 3 & 2 & 3 \\ 2 & 0 & 2\end{array}\right]=\left[\begin{array}{lll}3 & 0 & 1\end{array}\right]\) then \(2 p+q-r\) equals: \(\quad\) [Online April 22, 2013] (a) \(-3\) (b) \(-1\) (c) 4 (d) 2

The matrix \(A^{2}+4 A-5 I\), where \(I\) is identity matrix and \(A=\left[\begin{array}{cc}1 & 2 \\ 4 & -3\end{array}\right]\), equals [Online April 9, 2013] (a) \(4\left[\begin{array}{ll}2 & 1 \\ 2 & 0\end{array}\right]\) (b) \(4\left[\begin{array}{cc}0 & -1 \\ 2 & 2\end{array}\right]\) (c) \(32\left[\begin{array}{ll}2 & 1 \\ 2 & 0\end{array}\right]\) (d) \(32\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]\)

If \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 2 & 1 & 0 \\ -3 & 2 & 1\end{array}\right]\) and \(B=\left[\begin{array}{ccc}1 & 0 & 0 \\ -2 & 1 & 0 \\\ 7 & -2 & 1\end{array}\right]\) then \(A B\) equals \(\quad\) [Online May 26, 2012] (a) \(I\) (b) \(A\) (c) \(B\) (d) 0

If \(\omega \neq 1\) is the complex cube root of unity and matrix \(H=\left[\begin{array}{cc}\omega & 0 \\ 0 & \omega\end{array}\right]\), then \(\mathrm{H}^{70}\) is equal to \(\quad\) [2011RS] (a) 0 (b) \(-\mathrm{H}\) (c) \(\mathrm{H}^{2}\) (d) \(\mathrm{H}\)

The number of \(3 \times 3\) non-singular matrices, with four entries as 1 and all other entries as 0, is (a) 5 (b) 6 (c) at least 7 (d) less than 4

Let \(A=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)\) and \(B=\left(\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right), a, b \in N\). Then [2006] (a) there cannot exist any B such that \(\mathrm{AB}=\mathrm{BA}\) (b) there exist more than one but finite number of \(\mathrm{B}^{\prime} \mathrm{s}\) such that \(\mathrm{AB}=\mathrm{BA}\) (c) there exists exactly one \(\mathrm{B}\) such that \(\mathrm{AB}=\mathrm{BA}\) (d) there exist infinitely many B's such that \(\mathrm{AB}=\mathrm{BA}\)

If \(A\) and \(B\) are square matrices of size \(n \times n\) such that \(A^{2}-B^{2}=(A-B)(A+B)\), then which of the following will be always true? (a) \(A=B\) (b) \(A B=B A\) (c) either of \(A\) or \(B\) is a zero matrix (d) either of \(A\) or \(B\) is identity matrix

If \(A=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\) and \(I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\), then which one of the following holds for all \(n \geq 1\), by the principle of mathematical induction (a) \(A^{n}=n A-(n-1) \mathrm{I}\) (b) \(A^{n}=2^{n-1} A-(n-1) \mathrm{I}\) (c) \(A^{n}=n A+(n-1) \mathrm{I}\) (d) \(A^{n}=2^{n-1} A+(n-1) \mathrm{I}\)

If \(A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]\) and \(A^{2}=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]\), then [2003] (a) \(\alpha=2 a b, \beta=a^{2}+b^{2}\) (b) \(\alpha=a^{2}+b^{2}, \beta=a b\) (c) \(\alpha=a^{2}+b^{2}, \beta=2 a b\) (d) \(\alpha=a^{2}+b^{2}, \beta=a^{2}-b^{2}\).

Let \(a, b, c \in \mathbf{R}\) be all non-zero and satisfy \(a^{3}+b^{3}+c^{3}=2 .\) If the matrix \(A=\left(\begin{array}{lll}a & b & c \\\ b & c & a \\ c & a & b\end{array}\right)\) satisfies \(A^{T} A=I\), then a value of \(a b c\) can be : [Sep. \(\mathbf{0 2}, \mathbf{2 0 2 0}\) (II)] (a) \(-\frac{1}{3}\) (b) \(\frac{1}{3}\) (c) 3 (d) \(\frac{2}{3}\)

The number of all \(3 \times 3\) matrices \(A\), with enteries from the set \(\\{-1,0,1\\}\) such that the sum of the diagonal elements of \(A A^{T}\) is 3, is

If \(\mathrm{A}=\left(\begin{array}{ll}2 & 2 \\ 9 & 4\end{array}\right)\) and \(\mathrm{I}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\), then \(10 \mathrm{~A}^{-1}\) is equal to: [Jan. 8, 2020 (II)] (a) \(A-4 l\) (b) \(6 I-A\) (c) \(A-6 I\) (d) \(4 I-A\)

If A is a symmetric matrix and \(\mathrm{B}\) is a skew-symmetrix matrix such that \(\mathrm{A}+\mathrm{B}=\left[\begin{array}{cc}2 & 3 \\ 5 & -1\end{array}\right]\), then \(\mathrm{AB}\) is equal to : [April 12, 2019 (I)] (a) \(\left[\begin{array}{cc}-4 & -1 \\ -1 & 4\end{array}\right]\) (b) \(\left[\begin{array}{cc}4 & -2 \\ -1 & -4\end{array}\right]\) (c) \(\left[\begin{array}{ll}4 & -2 \\ 1 & -4\end{array}\right]\) (d) \(\left[\begin{array}{cc}-4 & 2 \\ 1 & 4\end{array}\right]\)

The total number of matrices \(A=\left(\begin{array}{ccc}0 & 2 y & 1 \\ 2 x & y & -1 \\ 2 x & -y & 1\end{array}\right),(x, y \in\) \(\mathrm{R}, x \neq y\) ) for which \(\mathrm{A}^{\mathrm{T}} \mathrm{A}=3 \mathrm{I}_{3}\) is: \(\quad\) [April 09, 2019 (II)] (a) 2 (b) 3 (c) 6 (d) 4

Let \(A=\left(\begin{array}{ccc}0 & 2 \mathrm{q} & r \\ p & q & -r \\ p & -q & r\end{array}\right)\). If \(\mathrm{AA}^{\mathrm{T}}=\mathrm{I}_{3}\), then \(|\mathrm{p}|\) is : [Jan. 11, 2019 (I)] (a) \(\frac{1}{\sqrt{5}}\) (b) \(\frac{1}{\sqrt{3}}\) (c) \(\frac{1}{\sqrt{2}}\) (d) \(\frac{1}{\sqrt{6}}\)

For two \(3 \times 3\) matrices \(\mathrm{A}\) and \(\mathrm{B}\), let \(\mathrm{A}+\mathrm{B}=2 \mathrm{~B}^{\mathrm{T}}\) and \(3 \mathrm{~A}+2 \mathrm{~B}=\) \(\mathrm{I}_{3}\), where \(\mathrm{B}^{\mathrm{T}}\) is the transpose of \(\mathrm{B}\) and \(\mathrm{I}_{3}\) is \(3 \times 3\) identity matrix. Then: \(\quad\) [Online April 9, 2017] (a) \(5 \mathrm{~A}+10 \mathrm{~B}=2 \mathrm{I}_{3}\) (b) \(10 \mathrm{~A}+5 \mathrm{~B}=3 \mathrm{I}_{3}\) (c) \(\mathrm{B}+2 \mathrm{~A}=\mathrm{I}_{3}\) (d) \(3 \mathrm{~A}+6 \mathrm{~B}=2 \mathrm{I}_{3}\)

If \(P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\) and \(\mathrm{Q}=\mathrm{PAP}^{\mathrm{T}}\), then \(\mathrm{P}^{\mathrm{T}} \mathrm{Q}^{2015} \mathrm{P}\) is ; \(\quad\) [Online April 9, 2016] (a) \(\left[\begin{array}{cc}0 & 2015 \\ 0 & 0\end{array}\right]\) (b) \(\left[\begin{array}{cc}2015 & 0 \\ 1 & 2015\end{array}\right]\) (c) \(\left[\begin{array}{cc}1 & 2015 \\ 0 & 1\end{array}\right]\) (d) \(\left[\begin{array}{cc}2015 & 1 \\ 0 & 2015\end{array}\right]\)

If \(\mathrm{A}=\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & -2 \\ \mathrm{a} & 2 & \mathrm{~b}\end{array}\right]\) is a matrix satisfying the equation \(\mathrm{AA}^{\mathrm{T}}=9 \mathrm{I}\), where \(\mathrm{I}\) is \(3 \times 3\) identity matrix, then the ordered pair (a, b) is equal to: (a) \((2,1)\) (b) \((-2,-1)\) (c) \((2,-1)\) (d) \((-2,1)\)

Let \(\mathrm{A}\) and \(\mathrm{B}\) be any two \(3 \times 3\) matrices. If \(\mathrm{A}\) is symmetric and \(\mathrm{B}\) is skewsymmetric, then the matrix \(\mathrm{AB}-\mathrm{BA}\) is: (a) skewsymmetric [Online April 19, 2014] (b) symmetric (c) neither symmetric nor skewsymmetric (d) I or \(-\mathrm{I}\), where \(\mathrm{I}\) is an identity matrix.

If \(A=\left(\begin{array}{c}\alpha-1 \\ 0 \\ 0\end{array}\right), B=\left(\begin{array}{c}\alpha+1 \\ 0 \\ 0\end{array}\right)\) be two matrices, then \(A B^{T}\) is a non-zero matrix for \(|\alpha|\) not equal to \(\quad\) [Online May 7, 2012] (a) 2 (b) 0 (c) 1 (d) 3

Let \(A\) and \(B\) be two symmetric matrices of order 3 . [2011] Statement-1: \(A(B A)\) and \((A B) A\) are symmetric matrices. Statement- \(\mathbf{2}: A B\) is symmetric matrix if matrix multiplication of \(A\) with \(B\) is commutative. (a) Statement- 1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement- 1 . (b) Statement- 1 is true, Statement- 2 is false. (c) Statement- 1 is false, Statement- 2 is true. (d) Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation for Statement-1.