Matrices; algebra of matrices; types of matrices — practice questions

Question 1 · difficulty L2 · understanding

The number of symmetric matrices of order 3, with all the entries from the set $$\{0,1,2,3,4,5,6,7,8,9\}$$ is :

• A. $$10^{9}$$
• B. $$9^{10}$$
• C. $$10^{6}$$
• D. $$6^{10}$$

Question 2 · difficulty L2 · understanding

$$\text { Let } A=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right] \text { and } B=\left[\begin{array}{ccc} 9^{2} & -10^{2} & 11^{2} \\ 12^{2} & 13^{2} & -14^{2} \\ -15^{2} & 16^{2} & 17^{2} \end{array}\right] \text {, then the value of } A^{\prime} B A \text { is: }$$

• A. 1224
• B. 1042
• C. 540
• D. 539

Question 3 · difficulty L2 · understanding

If \begin{aligned}P = \left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right]\end{aligned}, then P 50 is :

• A. \begin{aligned}\left[ {\matrix{ 1 & 0 \cr {25} & 1 \cr } } \right]\end{aligned}
• B. \begin{aligned}\left[ {\matrix{ 1 & {50} \cr 0 & 1 \cr } } \right]\end{aligned}
• C. \begin{aligned}\left[ {\matrix{ 1 & {25} \cr 0 & 1 \cr } } \right]\end{aligned}
• D. \begin{aligned}\left[ {\matrix{ 1 & 0 \cr {50} & 1 \cr } } \right]\end{aligned}

Question 4 · difficulty L2 · understanding

Let \begin{aligned}A + 2B = \left[ {\matrix{ 1 & 2 & 0 \cr 6 & { - 3} & 3 \cr { - 5} & 3 & 1 \cr } } \right]\end{aligned} and \begin{aligned}2A - B = \left[ {\matrix{ 2 & { - 1} & 5 \cr 2 & { - 1} & 6 \cr 0 & 1 & 2 \cr } } \right]\end{aligned}. If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A) $$-$$ Tr(B) has value equal to

• A. 1
• B. 2
• C. 0
• D. 3

Question 5 · difficulty L2 · understanding

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A 2 is 1, then the possible number of such matrices is :

• A. 6
• B. 4
• C. 1
• D. 12

Question 6 · difficulty L2 · understanding

If for the matrix, \begin{aligned}A = \left[ {\matrix{ 1 & { - \alpha } \cr \alpha & \beta \cr } } \right]\end{aligned}, $$A{A^T} = {I_2}$$, then the value of $${\alpha ^4} + {\beta ^4}$$ is :

• A. 3
• B. 2
• C. 1
• D. 4

Question 7 · difficulty L2 · understanding

Let $$A = \left[ {\matrix{ {{a_1}} \cr {{a_2}} \cr } } \right]$$ and $$B = \left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr } } \right]$$ be two 2 $$\times$$ 1 matrices with real entries such that A = XB, where \begin{aligned}X = {1 \over {\sqrt 3 }}\left[ {\matrix{ 1 & { - 1} \cr 1 & k \cr } } \right]\end{aligned}, and k$$\in$$R. If $$a_1^2$$ + $$a_2^2$$ = $${2 \over 3}$$(b$$_1^2$$ + b$$_2^2$$) and (k 2 + 1) b$$_2^2$$ $$\ne$$ $$-$$2b 1 b 2 , then the value of k is __________.

Question 8 · difficulty L2 · understanding

Which one of the following matrices can be obtained by performing elementary row transformations on the $3 \times 3$ identity matrix?

• A. $$\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$$
• B. $$\begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 1 \end{bmatrix}$$
• C. $$\begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 2 & 5 & 8 \end{bmatrix}$$
• D. $$\begin{bmatrix} 1 & 1 & 1 \\ -1 & 1 & 2 \\ 0 & 2 & 3 \end{bmatrix}$$

Question 9 · difficulty L2 · understanding

If P is a 3 $$\times$$ 3 matrix such that P T = 2P + I, where P T is the transpose of P and I is the 3 $$\times$$ 3 identity matrix, then there exists a column matrix $$X = \left[ {\matrix{ x \cr y \cr z \cr } } \right] \ne \left[ {\matrix{ 0 \cr 0 \cr 0 \cr } } \right]$$ such that

• A. $$PX = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr } } \right]$$
• B. PX = X
• C. PX = 2X
• D. PX = $$-$$X

Question 10 · difficulty L3 · understanding

Let $\mathrm{A}=\left[$$\begin{array}{lll}1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1\end{array}$$\right]$and$\mathrm{B}=\left[\mathrm{b}_{i j}\right], 1 \leq i, j \leq 3$. If$\mathrm{B}=\mathrm{A}^{99}-\mathrm{I}$, then the value of$\frac{\mathrm{b}_{31}-\mathrm{b}_{21}}{\mathrm{~b}_{32}}$ is :

• A. 99
• B. 199
• C. 149
• D. 159