$\mathrm{What} \mathrm{is} \mathrm{the} \mathrm{order} \mathrm{of} \left[\begin{array}{ccc}\mathrm{x}& \mathrm{y}& \mathrm{z}\end{array}\right]\left[\begin{array}{ccc}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{h}& \mathrm{b}& \mathrm{f}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}\end{array}\right]\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right] = ?$

• $\mathrm{3 \times 1}$

• $1 \times 1$

• $1 \times 3$

• $3 \times 3$

$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right], \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} {\mathrm{A}}^4 \mathrm{is}$

• $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$

• $\left[\begin{array}{cc}0& 0\\ 1& 1\end{array}\right]$

• $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$$\begin{gathered} \mathrm{A} = \left[\begin{array}{cc}\mathrm{x} + \mathrm{y}& \mathrm{y}\\ \mathrm{x}& \mathrm{x} - \mathrm{y}\end{array}\right], \mathrm{B} = \left[\begin{array}{c}3\\ - 2\end{array}\right] \mathrm{and} \mathrm{C} = \left[\begin{array}{c}4\\ - 2\end{array}\right].\mathrm{If} \mathrm{AB} = \mathrm{C}, \\ \mathrm{then} \mathrm{what} \mathrm{is} {\mathrm{A}}^2 = ? \end{gathered}$$

• $\left[\begin{array}{cc}4& 8\\ - 4& - 16\end{array}\right]$

• $\left[\begin{array}{cc}4& - 4\\ 8& - 16\end{array}\right]$

• $\left[\begin{array}{cc} - 4& - 8\\ 4& 12\end{array}\right]$

• $\left[\begin{array}{cc} - 4& - 8\\ 8& 12\end{array}\right]$

The matrix A has x rows and x + 5 columns. The matrix B has y rows and 11 - y columns. Both AB and BA exist. What are the values of x and y respectively ?

• 8 and 3

• 3 and 4

• 3 and 8

• 8 and 8

$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}4\mathrm{i} - 6& 10\mathrm{i}\\ 14\mathrm{i}& 6 + 4\mathrm{i}\end{array}\right] \mathrm{and} \mathrm{k} = \frac{1}{2\mathrm{i}}, \mathrm{where} \mathrm{i} = \sqrt{ - 1}, \mathrm{then} \mathrm{kA} = ?$

• $\left[\begin{array}{cc}2 + 3\mathrm{i}& 5\\ 7& 6 + 4\mathrm{i}\end{array}\right]$

• $\left[\begin{array}{cc}2 - 3\mathrm{i}& 5\\ 7& 2 + 3\mathrm{i}\end{array}\right]$

• $\left[\begin{array}{cc}2 - 3\mathrm{i}& 7\\ 5& 2 + 3\mathrm{i}\end{array}\right]$

• $\left[\begin{array}{cc}2 + 3\mathrm{i}& 5\\ 7& 2 \hspace{0.17em}+3\mathrm{i}\end{array}\right]$

A is a square matrix, then the value of adj A^T - adj(A)^T is equal to

• A

• 2 $\left|\mathrm{A}\right|$ I, where the identity matrix

• Null matrix whose order is same as that of A

• unit matrix whose order is same as that of A

If a, b, c are non-zero real numbers, then the inverse of the matrix A = $\left[\begin{array}{ccc}\mathrm{a}& 0& 0\\ 0& \mathrm{b}& 0\\ 0& 0& \mathrm{c}\end{array}\right] =\hspace{0.17em}?$

• $\left[\begin{array}{ccc}{\mathrm{a}}^{ - 1}& 0& 0\\ 0& {\mathrm{b}}^{ - 1}& 0\\ 0& 0& {\mathrm{c}}^{ - 1}\end{array}\right]$

• $\frac{1}{\mathrm{abc}}\left[\begin{array}{ccc}{\mathrm{a}}^{ - 1}& 0& 0\\ 0& {\mathrm{b}}^{ - 1}& 0\\ 0& 0& {\mathrm{c}}^{ - 1}\end{array}\right]$

• $\frac{1}{\mathrm{abc}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

• $\frac{1}{\mathrm{abc}}\left[\begin{array}{ccc}\mathrm{a}& 0& 0\\ 0& \mathrm{b}& 0\\ 0& 0& \mathrm{c}\end{array}\right]$

If A = $\left(\begin{array}{cc} - 2& 2\\ 2& - 2\end{array}\right)$, then which one of the following is correct ?

• A² = - 2A

• A² = - 4A

• A² = - 3A

• A² = 4A

29.

If A is a square matrix, then what is adj(A ^{- 1}) - (adj A) ^{- 1} equal to ?

• 2$\left|\mathrm{A}\right|$

• Null matrix

• Unit Matrix

• None of the above

Consider the following in respect of the matrix

$$\begin{gathered} \mathrm{A} = \left(\begin{array}{cc} - 1& 1\\ 1& - 1\end{array}\right) \\ 1. {\mathrm{A}}^2 = - \mathrm{A} \\ 2. {\mathrm{A}}^3 = 4\mathrm{A} \end{gathered}$$

Which of the above is/are correct ?

• 1 only

• 2 only

• bth 1 and 2

• neither 1 nor 2