## Matrices and Determinants Questions and Answers Part-9

1.If $D_{1}$ and $D_{2}$ are two $3 \times3$  diagonal matrices, then
a) $D_{1}D_{2}$  is diagonal matrix
b) $D_{1}D_{2}=D_{2}D_{1}$
c) $D_1^2+D_2^2$  is a diagonal matrix
d) All of the above

Explanation:

2. Let $A=\begin{bmatrix}1 & 0 \\1 & 1 \end{bmatrix}$   then
a) $A^{-n}=\begin{bmatrix}1 & 0 \\-n & 1 \end{bmatrix}\forall n \epsilon N$
b) $\lim_{n \rightarrow \infty}\frac{1}{n^{2}}A^{-n}=\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix}$
c) $\lim_{n \rightarrow \infty}\frac{1}{n}A^{-n}=\begin{bmatrix}0 & 0 \\-1 & 0 \end{bmatrix}$
d) All of the above

Explanation:

3. Let A, B and C be $2 \times2$  matrices with entries from the set of real numbers. Define '*' as follows
$A*B=\frac{1}{2}\left(AB+BA\right)$     , then
a) A * B = B * A
b) $A * A = A^{2}$
c) A * (B + C) = A * B + A * C
d) All of the above

Explanation: A * B = B * A

4. With A, B, C as in Q.No.3, define 'o' as follows: $AoB=\frac{1}{2}\left(AB-BA\right)$     , then
a) AoA = O
b) AoI = O
c) Ao(B + C) = AoB + AoC
d) All of the above

Explanation: AoA = O

5. Suppose a, b > 0 and let $\triangle\left(x\right)=\begin{bmatrix}x & a & b \\a & x & b \\a & b & x\end{bmatrix},x \epsilon R$
then $\frac{\triangle'\left(0\right)}{\triangle\left(0\right)}$   is equal to
a) $\frac{1}{a+b}$
b) $1-\left(\frac{1}{a}+\frac{1}{b}\right)$
c) 0
d) $\frac{1}{a+b}-\frac{1}{a}-\frac{1}{b}$

Explanation:

6. Let $\triangle=\begin{bmatrix}ln \left(2/15\right) & ln\left(4\right) & ln\left(4\right) \\ ln\left(9\right) & ln\left(3/10\right) & ln\left(9\right) \\ ln\left(25\right) & ln\left(25\right) & ln\left(5/6\right)\end{bmatrix}$
then $\triangle$ is equal to
a) $\left[ln\left(30\right)\right]^{3}$
b) $\left[ln\left(60\right)\right]^{3}$
c) $\left[ln\left(30\right)\right]^{2}$
d) $\left[ln\left(60\right)\right]^{2}$

Explanation: Let a = ln(2), b = ln(3), c = ln(5), then

7. Let $\triangle$ ABC be an acute angled triangle such that
$\frac{\tan A}{\tan B}-\frac{\tan B}{\tan A}+\frac{\tan B}{\tan C}-\frac{\tan C}{\tan B}+\frac{\tan C}{\tan A}-\frac{\tan A}{\tan C}=0$
then
a) $\triangle ABC$   is an equilateral triangle
b) $\triangle ABC$   is an isosceles triangle
c) one of the angles of $\triangle ABC$   is $\pi/3$
d) one of the angle of $\triangle ABC$   must be $\pi/4$

Explanation: Let x = tan A, y = tan B, z = tan C, then

8. Let a, b, c be distinct positive real numbers such that $S=\sum\frac{1}{a\left(a-b\right)\left(a-c\right)}$     and $T=\frac{1}{abc}$
then
a) S=T
b) S+T=0
c) S = (a + b + c)T
d) S = (bc + ca + ab)T

Explanation:

9. Let x, y, z be three real distinct real numbers, and let
$S=\begin{bmatrix}x+y^{2} & y+z^{2} & z+x^{2} \\x^{2}+y^{3} & y^{2}+z^{3} & z^{2}+x^{3} \\x^{3}+y^{4} & y^{3}+z^{4} & z^{3}+x^{4}\end{bmatrix}$
and $T=\begin{bmatrix}1 & 1 & 1 \\x & y & z \\x^{2} & y^{2} & z^{2}\end{bmatrix}$
then
a) $\frac{\triangle}{\triangle_{1}}=xyz$
b) $\frac{\triangle_{1}}{\triangle}=\frac{1}{1+x^{2}y^{2}z^{2}}$
c) $\frac{\triangle_{1}}{\triangle}=\frac{1}{1+xyz}$
d) $\frac{\triangle_{1}}{\triangle}=xyz +x^{2}y^{2}z^{2}$

10. Let $A=\left(a_{ij}\right)_{3\times 3}$     be such that det(A) = 5. Suppose $b_{ij}=2^{i+j}a_{ij}\left(1\leq i, j\leq3\right)$       and let $B=\left(b_{ij}\right)_{3 \times 3}$     , then det(B) is equal to
a) $2^{16}$
b) $2^{15}$
c) $2^{14}$
d) $2^{12}$