Chapter 19

Let \(\mathrm{A}\) and \(\mathrm{B}\) be two invertible matrices of order \(3 \times 3\). If \(\operatorname{det}\left(\mathrm{ABA}^{\mathrm{T}}\right)=8\) and \(\operatorname{det}\left(\mathrm{AB}^{-1}\right)=8\), then \(\operatorname{det}\left(\mathrm{BA}^{-1} \mathrm{~B}^{\mathrm{T}}\right)\) is equal to: [Jan. 11, 2019 (II)] (a) \(\frac{1}{4}\) (b) 1 (c) \(\frac{1}{16}\) (d) 16

If \(\mathrm{A}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]\), then the matrix \(\mathrm{A}^{-50}\) when \(\theta=\frac{\pi}{12}\), is equal to: \(\quad\) Jan 09, 2019 (I)] (a) \(\left[\begin{array}{cc}\frac{1}{2} & -\frac{\sqrt{3}}{2} \\\ \frac{\sqrt{3}}{2} & \frac{1}{2}\end{array}\right]\) (b) \(\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right]\) (c) \(\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right]\) (d) \(\left[\begin{array}{cc}\frac{1}{2} & \frac{\sqrt{3}}{2} \\\ -\frac{\sqrt{3}}{2} & \frac{1}{2}\end{array}\right]\)

Suppose \(A\) is any \(3 \times 3\) non-singular matrix and \((A-3 I)(A-5 I)=O\), where \(l=I_{3}\) and \(O=O_{3}\). If \(\alpha A+\beta A^{-1}=4 I\), then \(\alpha+\beta\) is equal to [Online April 15, 2018] (a) 8 (b) 12 (c) 13 (d) 7

If \(\mathrm{A}=\left[\begin{array}{cc}2 & -3 \\ -4 & 1\end{array}\right]\), then \(\operatorname{adj}\left(3 \mathrm{~A}^{2}+12 \mathrm{~A}\right)\) is equal to: \([\mathbf{2 0 1} 7]\) (a) \(\left[\begin{array}{cc}72 & -63 \\ -84 & 51\end{array}\right]\) (b) \(\left[\begin{array}{cc}72 & -84 \\ -63 & 51\end{array}\right]\) (c) \(\left[\begin{array}{ll}51 & 63 \\ 84 & 72\end{array}\right]\) (d) \(\left[\begin{array}{ll}51 & 84 \\ 63 & 72\end{array}\right]\)

Let A be any \(3 \times 3\) invertible matrix. Then which one of the following is not always true? \(\quad\) [Online April 8, 2017] (a) \(\operatorname{adj}(\mathrm{A})=|\mathrm{A}| \cdot \mathrm{A}^{-1}\) (b) \(\operatorname{adj}(\operatorname{adj}(\mathrm{A}))=|\mathrm{A}| \cdot \mathrm{A}\) (c) \(a d j(a d j(A))=|A|^{2} \cdot(a d j(A))^{-1}\) (d) \(\operatorname{adj}(\operatorname{adj}(\mathrm{A}))=|\mathrm{A}| \cdot(\operatorname{adj}(\mathrm{A}))^{-1}\)

If \(\mathrm{A}=\left[\begin{array}{cc}5 \mathrm{a} & -\mathrm{b} \\ 3 & 2\end{array}\right]\) and \(\mathrm{A} \mathrm{adj} \mathrm{A}=\mathrm{A} \mathrm{A}^{\mathrm{T}}\), then \(5 \mathrm{a}+\mathrm{b}\) is equal to: [2016] (a) 4 (b) 13 (c) \(-1\) (d) 5

Let \(\mathrm{A}\) be a \(3 \times 3\) matrix such that \(\mathrm{A}^{2}-5 \mathrm{~A}+7 \mathrm{I}=0\). Statement \(-\mathrm{I}: \mathrm{A}^{-1}=\frac{1}{7}(5 \mathrm{I}-\mathrm{A})\) Statement II : the polynomial \(\mathrm{A}^{3}-2 \mathrm{~A}^{2}-3 \mathrm{~A}+\alpha\) can be reduced to \(5(\mathrm{~A}-4 \mathrm{I})\). [Online April 10, 2016] Then: (a) Both the statements are true. (b) Both the statements are false. (c) Statement-I is true, but Statement-II is false. (d) Statement I is false, but Statement-II is true.

If \(\mathrm{A}\) is a \(3 \times 3\) matrix such that \(|5 \cdot \mathrm{adj} \mathrm{A}|=5\), then \(|\mathrm{A}|\) is equal to : [Online April 11, 2015] (a) \(\pm \frac{1}{5}\) (b) \(\pm \frac{1}{25}\) (c) \(\pm 1\) (d) \(\pm 5\)

If \(A\) is an \(3 \times 3\) non-singular matrix such that \(A A^{\prime}=A^{\prime} A\) and \(B=A^{-1} A^{\prime}\), then \(\mathrm{BB}^{\prime}\) equals: (a) \(B^{-1}\) (b) \(\left(B^{-1}\right)^{\prime}\) (c) \(I+B\) (d) \(I\)

Let \(\mathrm{A}\) be a \(3 \times 3\) matrix such that \(A\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]\) \(\begin{array}{ll}\text { Then } \mathrm{A}^{-1} \text { is: } & \text { [Online April 11, 2014] }\end{array}\) (a) \(\left[\begin{array}{lll}3 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 1\end{array}\right]\) (b) \(\left[\begin{array}{lll}3 & 2 & 1 \\ 3 & 2 & 0 \\ 1 & 1 & 0\end{array}\right]\) (c) \(\left[\begin{array}{lll}0 & 1 & 3 \\ 0 & 2 & 3 \\ 1 & 1 & 1\end{array}\right]\) (d) \(\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 1 \\ 0 & 2 & 3\end{array}\right]\)

If \(P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]\) is the adjoint of a \(3 \times 3\) matrix \(\mathrm{A}\) and \(|\mathrm{A}|=4\), then \(\alpha\) is equal to : (a) 4 (b) 11 (c) 5 (d) 0

Let \(P\) and \(Q\) be \(3 \times 3\) matrices \(P \neq Q .\) If \(P^{3}=Q^{3}\) and \(P^{2} Q=Q^{2} P\) then determinant of \(\left(P^{2}+Q^{2}\right)\) is equal to : (a) \(-2\) (b) 1 [2012] (c) 0 (d) \(-1\)

Let \(A=\left(\begin{array}{lll}1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1\end{array}\right)\). 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 equal to: \([2012]\) (a) \(\left(\begin{array}{c}-1 \\ 1 \\ 0\end{array}\right)\) (b) \(\left(\begin{array}{c}-1 \\ 1 \\ -1\end{array}\right)\) (c) \(\left(\begin{array}{c}-1 \\ -1 \\ 0\end{array}\right)\) (d) \(\left(\begin{array}{c}1 \\ -1 \\ -1\end{array}\right)\)

If \(A^{T}\) denotes the transpose of the matrix \(A=\left[\begin{array}{lll}0 & 0 & a \\ 0 & b & c \\ d & e & f\end{array}\right]\), where \(a, b, c, d, e\) and \(f\) are integers such that \(a b d \neq 0\), then the number of such matrices for which \(A^{-1}=A^{T}\) is [Online May 19, 2012] (a) 2(3!) (b) \(3(2 !)\) (c) \(2^{3}\) (d) \(3^{2}\)

Let \(A\) and \(B\) be real matrices of the form \(\left[\begin{array}{ll}\alpha & 0 \\ 0 & \beta\end{array}\right]\) and \(\left[\begin{array}{ll}0 & \gamma \\\ \delta & 0\end{array}\right]\), respectively. \(\quad\) Online May 12, 2012] Statement 1: \(A B-B A\) is always an invertible matrix. Statement 2: \(A B-B A\) is never an identity matrix. (a) Statement 1 is true, Statement 2 is false. (b) Statement 1 is false, Statement 2 is true. (c) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1 . (d) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1 .

Consider the following relation \(\mathrm{R}\) on the set of real square matrices of order \(3 . \quad[2011 \mathrm{RS}]\) \(R=\left\\{(A, B) \mid A=P^{-1} B P\right.\) for some invertible matrix \(\left.P\right\\}\) Statement- \(\mathbf{1}: R\) is equivalence relation. Statement- \(2:\) For any two invertible \(3 \times 3\) matrices \(M\) and \(N,(M N)^{-1}=N^{-1} M^{-1}\) (a) Statement- 1 is true, statement- 2 is true and statement2 is a correct explanation for statement-1. (b) Statement- 1 is true, statement-2 is true; statement- 2 is not a correct explanation for statement-1. (c) Statement- 1 is true, stement- 2 is false. (d) Statement- 1 is false, statement- 2 is true.

Let A be a \(2 \times 2\) matrix Statement \(-1: \operatorname{adj}(\operatorname{adj} \mathrm{A})=\mathrm{A}\) Statement \(-\mathbf{2}: \mid\) adj \(\mathrm{A}|=| \mathrm{A} \mid \quad\) [2009] (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.

Let \(A\) be a square matrix all of whose entries are integers. Then which one of the following is true? [2008] (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 but all its entries are integers (d) If det \(A=\pm 1\), then \(A^{-1}\) need not exists

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

Let \(A=\left(\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right)\) and \(B=\left(\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3\end{array}\right)\). If \(B\) is the inverse of matrix \(A\), then \(\alpha\) is (a) 5 (b) \(-1\) (c) 2 (d) \(-2\)

Let \(A=\left(\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right)\). The only correct statement about the matrix \(A\) is (a) \(A^{2}=I\) (b) \(A=(-1) I\), where \(I\) is a unit matrix (c) \(A^{-1}\) does not exist (d) \(A\) is a zero matrix

The values of \(\lambda\) and \(\mu\) for which the system of linear equations [Sep. 06, 2020 (I)] \(x+y+z=2\) \(x+2 y+3 z=5\) \(x+3 y+\lambda z=\mu\) has infinitely many solutions are, respectively: (a) 6 and 8 (b) 5 and 7 (c) 5 and 8 (d) 4 and 9

The sum of distinct values of \(\lambda\) for wheih the system of equations \((\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0\) \((\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0\) \(2 x+(3 \lambda+1) y+3(\lambda-1) z=0\), has non-zero solutions, is [NA Sep. 06, 2020 (II)]

Let \(\lambda \in \mathrm{R}\). The system of linear equations \(2 x_{1}-4 x_{2}+\lambda x_{3}=1\) [Sep. 05, 2020 (I)] \(x_{1}-6 x_{2}+x_{3}=2\) \(\lambda x_{1}-10 x_{2}+4 x_{3}=3\) (a) exactly one negative value of \(\lambda\) (b) exactly one positive value of \(\lambda\) (c) every value of \(\lambda\) (d) exactly two value of \(\lambda\)

If the system of linear equations \(x+y+3 z=0\) \(x+3 y+k^{2} z=0\) \(3 x+y+3 z=0\) has a non-zero solution \((x, y, z)\) for some \(k \in \mathbf{R}\), then \(x+\left(\frac{y}{z}\right)\) is equal to: \(\quad\) [Sep. 05, 2020 (II)] (a) \(-3\) (b) 9 (c) 3 (d) \(-9\)

If the system of equations \(x-2 y+3 z=9,2 x+y+z=b\) \(x-7 y+a z=24\), has infinitely many solutions, then \(a-b\) is equal to [NA Sep. 04, 2020 (I)]

Suppose the vectors \(x_{1}, x_{2}\) and \(x_{3}\) are the solutions of the system of linear equations, \(A x=b\) when the vector \(b\) on the right side is equal to \(b_{1}, b_{2}\) and \(b_{3}\) respectively. If \(x_{1}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x_{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x_{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b_{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right], b_{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]\) and \(b_{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right]\), then the determinant of \(A\) is equal to: [Sep. 04, 2020 (II)] (a) 4 (b) 2 (c) \(\frac{1}{2}\) (d) \(\frac{3}{2}\)

If the system of equations \(x+y+z=2\) \(2 x+4 y-z=6\) \(3 x+2 y+\lambda z=\mu\) has infinitely many solutions, then: [Sep. 04, 2020 (II)] (a) \(\lambda+2 \mu=14\) (b) \(2 \lambda-\mu=5\) (c) \(\lambda-2 \mu=-5\) (d) \(2 \lambda+\mu=14\)

Let \(S\) be the set of all integer solutions, \((x, y, z)\), of the system of equations \(x-2 y+5 z=0\) \(-2 x+4 y+z=0\) \(-7 x+14 y+9 z=0\) such that \(15 \leq x^{2}+y^{2}+z^{2} \leq 150\). Then, the number of elements in the set \(S\) is equal to

The following system of linear equations \(7 x+6 y-2 z=0\) \(3 x+4 y+2 z=0\) \(x-2 y-6 z=0\), has \(\quad\) [Jan. 9,2020 (II)] (a) infinitely many solutions, \((x, y, z)\) satisfying \(y=2 z\). (b) no solution. (c) infinitely many solutions, \((x, y, z)\) satisfying \(x=2 z\). (d) only the trivial solution.

For which of the following ordered pairs \((\mu, \delta)\), the system of linear equations \(x+2 y+3 z=1\) \(3 x+4 y+5 z=\mu\) \(4 x+4 y+4 z=\delta\) is inconsistent? \(\quad\) [Jan. 8, 2020 (I)] (a) \((4,3)\) (b) \((4,6)\) (c) \((1,0)\) (d) \((3,4)\)

The system of linear equations \(\lambda x+2 y+2 z=5\) \(2 \lambda x+3 y+5 z=8\) \(4 x+\lambda y+6 z=10\) has: [Jan.8, 2020 (II)] (a) no solution when \(\lambda=8\) (b) a unique solution when \(\lambda=-8\) (c) no solution when \(\lambda=2\) (d) infinitely many solutions when \(\lambda=2\)

If the system of linear equations \(2 x+2 a y+a z=0\) \(2 x+3 b y+b z=0\) \(2 x+4 c y+c z=0\) where \(a, b, c \in \boldsymbol{R}\) are non-zero and distinct; has a non-zero solution, then: \(\quad\) [Jan. 7, 2020 (I)] (a) \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) are in A.P. (b) \(a, b, c\) are in GP. (c) \(a+b+c=0\) (d) \(a, b, c\) are in A.P.

If the system of linear equations, \(x+y+z=6\) \(x+2 y+3 z=10\) \(3 x+2 y+\lambda z=\mu\) has more than two solutions, then \(\mu-\lambda^{2}\) is equal to

If the system of linear equations \(x+y+z=5\) \(x+2 y+2 z=6\) \(\mathrm{x}+3 \mathrm{y}+\lambda \mathrm{z}=\mu,(\lambda, \mu \in \mathbf{R})\), has infinitely many solutions, then the value of \(\lambda+\mu\) is : [April 10, 2019 (I)] (a) 12 (b) 9 (c) 7 (d) 10

Let \(\lambda\) be a real number for which the system of linear equations: \(x+y+z=6\) \(4 x+\lambda y-\lambda z=\lambda-2\) \(3 x+2 y-4 z=-5\) has infinitely many solutions. Then \(\lambda\) is a root of the quadratic equation: \(\quad\) [April 10, 2019 (II)] (a) \(\lambda^{2}+3 \lambda-4=0\) (b) \(\lambda^{2}-3 \lambda-4=0\) (c) \(\lambda^{2}+\lambda-6=0\) (d) \(\lambda^{2}-\lambda-6=0\)

If the system of equations \(2 x+3 y-z=0, x+k y-2 z=0\) and \(2 x-y+z=0\) has a non-trivial solution \((x, y, z)\), then \(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k\) is equal to: \(\quad\) [April 09, 2019 (II)] (a) \(\frac{3}{4}\) (b) \(\frac{1}{2}\) (c) \(-\frac{1}{4}\) (d) \(-4\)

The greatest value of \(c \in R\) for which the system of linear equations \(x-c y-c z=0 ; c x-y+c z=0 ; c x+c y-z=0\) has a non-trivial solution, is : \(\quad\) [April 08, 2019 (I)] (a) \(-1\) (b) \(\frac{1}{2}\) (c) 2 (d) 0

An ordered pair \((\alpha, \beta)\) for which the system of linear equations \((1+\alpha) x+\beta y+z=2\) \(a x+(1+\beta) y+z=3\) \(\alpha x+\beta y+2 z=2\) has a unique solution, is : (a) \((2,4)\) (b) \((-3,1)\) (c) \((-4,2)\) (d) \((1,-3)\)

The set of all values of \(\lambda\) for which the system of linear equations \(x-2 y-2 z=\lambda x\) \(x+2 y+z=\lambda y\) \(-x-y=\lambda 2\) has a non-trivial solution: \(\quad\) [Jan. 12, 2019 (II)] (a) is a singleton (b) contains exactly two elements (c) is an empty set (d) contains more than two elements

If the system of linear equations \(2 x+2 y+3 z=a\) \(3 x-y+5 z=b\) \(x-3 y+2 z=c\) where, \(a, b, c\) are non-zero real numbers, has more than one solution, then : (a) \(\mathrm{b}-\mathrm{c}+\mathrm{a}=0\) (b) \(b-c-a=0\) (c) \(a+b+c=0\) (d) \(b+c-a=0\)

The number of values of \(\theta \in(0, \pi)\) for which the system of linear equations $$ \begin{aligned} &x+3 y+7 z=0 \\ &-x+4 y+7 z=0 \\ &(\sin 3 \theta) x+(\cos 2 \theta) y+2 z=0 \end{aligned} $$ has a non-trivial solution, is: [Jan. 10, 2019 (II)] (a) three (b) two (c) four (d) one

If the system of equations \(\quad\) [Jan 10, 2019 (I)] \(x+y+z=5\) \(x+2 y+3 z=9\) \(x+3 y+\alpha z=\beta\) has infinitely many solutions, then \(\beta-\alpha\) equals: (a) 21 (b) 8 (c) 18 (d) 5

If the system of linear equations \(x+k y+3 z=0\) \(3 x+k y-2 z=0\) \(2 x+4 y-3 z=0\) has a non-zero solution \((\mathrm{x}, \mathrm{y}, \mathrm{z})\), then \(\frac{\mathrm{xz}}{\mathrm{y}^{2}}\) is equal to: [2018] (a) 10 (b) \(-30\) (c) 30 (d) \(-10\)

The number of values of \(k\) for which the system of linear equations, \((k+2) x+10 y=k, k x+(k+3) y=k-1\) has no solution, is \(\quad\) [Online April 16, 2018] (a) Infinitely many (b) 3 (c) 1 (d) 2

Let \(S\) be the set of all real values of \(k\) for which the system of linear equations \(x+y+z=2\) \(2 x+y-z=3\) \(3 x+2 y+k z=4\) has a unique solution. Then \(S\) is [Online April 15, 2018] (a) an empty set (b) equal to \(\mathrm{R}-\\{0\\}\) (c) equal to \(\\{0\\}\) (d) equal to \(\mathrm{R}\)

If the system of linear equations \(x+a y+z=3\) \(x+2 y+2 z=6\) \(x+5 y+3 z=\mathbf{b}\) has no solution, then [Online April 15, 2018] (a) \(a=1, b \neq 9\) (b) \(a \neq-1, b=9\) (c) \(a=-1, b=9\) (d) \(a=-1, b \neq 9\)

If \(\mathrm{S}\) is the set of distinct values of ' \(\mathrm{b}\) ' for which the following system of linear equations [2017] \(x+y+z=1\) \(x+a y+z=1\) \(a x+b y+z=0\) has no solution, then \(S\) is : (a) a singleton (b) an empty set (c) an infinite set (d) a finite set containing two or more elements

The number of real values of \(\lambda\) for which the system of linear equations \(2 x+4 y-\lambda z=0\) \(4 x+\lambda y+2 z=0\) \(\lambda x+2 y+2 z=0\) has infinitely many solutions, is: [Online April 8, 2017] (a) 0 (b) 1 (c) 2 (d) 3

The system of linear equations \(\mathrm{x}+\lambda \mathrm{y}-\mathrm{z}=0\) \(\lambda x-y-z=0\) \(x+y-\lambda z=0\) has a non-trivial solution for: (a) exactly two values of \(\lambda\). (b) exactly three values of \(\lambda\). (c) infinitely many values of \(\lambda\). (d) exactly one value of \(\lambda\).