Chapter 5

If \(A=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]\), the values of \(\alpha, \beta\) such that \((\alpha I+\) \(\beta A)^{2}=A^{2}\) arc (A) \(\pm \frac{1}{\sqrt{2}}, \pm \frac{1}{\sqrt{2}}\) (B) \(\pm \frac{1}{\sqrt{2}}, \mp \frac{1}{\sqrt{2}}\) (C) \(\pm \frac{i}{\sqrt{2}}, \pm \frac{i}{\sqrt{2}}\) (D) \(\pm \frac{i}{\sqrt{2}}, \mp \frac{i}{\sqrt{2}}\)

The rank of the matrix \(A=\left[\begin{array}{llll}0 & 1 & 2 & 1 \\ 1 & 2 & 3 & 2 \\ 3 & 1 & 1 & 3\end{array}\right]\) is (A) 1 (B) 2 (C) 3 (D) 4

Rank of the matrix \(A=\left[\begin{array}{rrrr}1 & -1 & 2 & -3 \\ 4 & 1 & 0 & 2 \\ 0 & 3 & 1 & 4 \\ 0 & 1 & 0 & 2\end{array}\right]\) is (A) 1 (B) 2 (C) 3 (D) 4

The rank of the matrix \(A=\left[\begin{array}{rrr}2 & 3 & 4 \\ 3 & 1 & 2 \\\ -1 & 2 & 2\end{array}\right]\) is (A) 1 (B) 2 (C) 3 (D) can't determine

The rank of the matrix \(A=\left[\begin{array}{rrrr}1 & 3 & 4 & 3 \\ 3 & 9 & 12 & 9 \\ -1 & -3 & -4 & -3\end{array}\right]\) is (A) 1 (B) 2 (C) 3 (D) 0

If \(A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]\) and \(A^{2}=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha\end{array}\right]\), then (A) \(\alpha=a^{2}+b^{2}, \beta=a b\) (B) \(\alpha=a^{2}+b^{2}, \beta=2 a b\) (C) \(\alpha=a^{2}+b^{2}, \beta=a^{2}-b^{2}\) (D) \(\alpha=2 a b, \beta=a^{2}+b^{2}\)

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) \(A\) is a zero matrix (B) \(A^{2}=I\) (C) \(A^{-1}\) does not exist (D) \(A=(-1) I\), where \(I\) is a unit matrix

Let \(A=\left(\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right)\) (10) \(B=\left(\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3\end{array}\right) .\) If \(B\) is the inverse of matrix \(A\), then \(\alpha\) is (A) \(-2\) (B) 5 (C) 2 (D) \(-1\)

If \(A^{2}-A+I=0\), then the inverse of \(A\) is (A) \(A+I\) (B) \(A\) (C) \(A-I\) (D) \(I-A\)

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 [2005] (A) \(A^{n}=n A-(n-1) I\) (B) \(A^{n}=2^{n-1} A-(n-1) I\) (C) \(A^{n}=n A+(n-1) I\) (D) \(A^{n}=2^{n-1} A+(n-1) I\)

If \(A\) and \(B\) are square matrices of order \(n \times n\) such that \(A^{2}-B^{2}=(A-B)(A+B)\), then which of the following will always be true? [2006] (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 an identity matrix

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 (A) there cannot exist any \(B\) such that \(A B=B A\) (B) there exist more than one but finite number of \(B\) 's such that \(A B=B A\) (C) there exists exactly one \(B\) such that \(A B=B A\) (D) there exist infinitely many \(B\) 's such that \(A B=B A\)

Let \(A=\left[\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\\ 0 & 0 & 5\end{array}\right]\), If \(\left|A^{2}\right|=25\) then \(|\alpha|\) equals (A) 5 (B) 1 (C) \(1 / 5\) (D) 5

The number of \(3 \times 3\) non-singular matrices, with four entries as 1 and all other entries as 0 , is [2010] (A) 5 (B) 6 (C) at least 7 (D) less than 4

Let \(\mathrm{A}\) and \(\mathrm{B}\) be two symmetric matrices of order \(3 .\) \([2011]\) Statement 1: \(A(B A)\) and \((A B) A\) are symmetric matrices. Statement 2: \(A B\) is symmetric matrix if matrix multiplication of \(A\) and \(B\) 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

The number of values of \(k\), for which the system of equations \((k+1) x+8 y=4 k\) \(k x+(k+3) y=3 k-1\) has no solution, is (A) 1 (B) 2 (C) 3 (D) infinite

If \(\mathrm{A}\) is a \(3 \times 3\) non-singular matrix such that \(A A^{\prime}=\) \(A^{\prime} A\) and \(B=A^{-1} A^{\prime}\), then \(B B^{\prime}\) equals \(\quad\) [2014] (A) \(\mathrm{I}+\mathrm{B}\) (B) I (C) \(B^{-1}\) (D) \(\left(B^{-1}\right)^{\prime}\)

If \(A=\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{array}\right]\) is a matrix satisfying the equation \(A A^{T}=9 I\), where \(I\) is \(3 \times 3\) identity matrix, then ordered pair \((a, b)\) is equal to [2015] (A) \((-2,1)\) (B) \((2,1)\) (C) \((-2,-1)\) (D) \((2,-1)\)