For non-singular square matrices A, B and C of the same order, (AB^-1C)^-1 is equal to :

• A^-1BC^-1

• C^-1B^-1A^-1

• CBA^-1

• C^-1BA^-1

Let A = $\left[\begin{array}{cc}{\cos }^2\left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right)\\ \cos \left(\mathrm{\theta }\right)\sin \left(\mathrm{\theta }\right)& {\sin }^2\left(\mathrm{\theta }\right)\end{array}\right]$ and B = $\left[\begin{array}{cc}{\cos }^2\left(\mathrm{\varphi }\right)& \sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\varphi }\right)\\ \cos \left(\mathrm{\varphi }\right)\sin \left(\mathrm{\varphi }\right)& {\sin }^2\left(\mathrm{\varphi }\right)\end{array}\right]$, then AB = 0 if :

• $\mathrm{\theta } = \mathrm{n\varphi }, \mathrm{n} = 0, 1, 2, ...$

• $\mathrm{\theta } + \mathrm{\varphi } = \mathrm{n\pi }, \mathrm{n} = 0, 1, 2, ...$

• $\mathrm{\theta } = \mathrm{\varphi } + \left(2\mathrm{n} + 1\right)\frac{\mathrm{\pi }}{2}, \mathrm{n} = 0, 1, 2, ...$

• $\mathrm{\theta } = \mathrm{\varphi } + \mathrm{n}\frac{\mathrm{\pi }}{2}, \mathrm{n} = 0, 1, 2, ...$

Let X = $\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\\ {\mathrm{x}}_3\end{array}\right], \mathrm{A} = \left[\begin{array}{ccc}1& - 1& 2\\ 2& 0& 1\\ 3& 2& 1\end{array}\right] \mathrm{and} \mathrm{B} = \left[\begin{array}{c}3\\ 1\\ 4\end{array}\right]$. If AX = B, then X is equal to

• $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$

• $\left[\begin{array}{c}- 1\\ - 2\\ 3\end{array}\right]$

• $\left[\begin{array}{c}- 1\\ - 2\\ - 3\end{array}\right]$

• $\left[\begin{array}{c}- 1\\ 2\\ 3\end{array}\right]$

Let A and B are two square matrices such that AB = A and BA = B, then A² equals to :

• B

• A

• I

• O

115.

If X = $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, then Xⁿ, for n $\in$ , is equal to :

• 2^{n - 1}X

• n²X

• nX

• 2^{n + 1}X

If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]\mathrm{A}\left[\begin{array}{cc}- 3& 2\\ 5& - 3\end{array}\right] = \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then the matrix A is equal to :

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

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

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

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

If a, b, c, d, e and f are in GP, then the value of $\left|\begin{array}{ccc}{\mathrm{a}}^2& {\mathrm{d}}^2& \mathrm{x}\\ {\mathrm{b}}^2& {\mathrm{e}}^2& \mathrm{y}\\ {\mathrm{c}}^2& {\mathrm{f}}^2& \mathrm{z}\end{array}\right|$ depends on :

• x and y

• x and z

• y and z

• independent of x, y and z

Let X = $\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]$, D = $\left[\begin{array}{c}3\\ 5\\ 11\end{array}\right]$ and A = $\left[\begin{array}{ccc}1& - 1& - 2\\ 2& 1& 1\\ 4& - 1& - 2\end{array}\right]$, if X = A^{- 1}D, then X is equal to :

• $\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]$

• $\left[\begin{array}{c}\frac{8}{3}\\ \frac{- 1}{3}\\ 0\end{array}\right]$

• $\left[\begin{array}{c}\frac{- 8}{3}\\ 1\\ 0\end{array}\right]$

• $\left[\begin{array}{c}\frac{8}{3}\\ \frac{1}{3}\\ - 1\end{array}\right]$

f(x) = $\left|\begin{array}{ccc}1& \mathrm{x}& \mathrm{x} + 1\\ 2\mathrm{x}& \mathrm{x}\left(\mathrm{x} - 1\right)& \left(\mathrm{x} + 1\right)\mathrm{x}\\ 3\mathrm{x}\left(\mathrm{x} - 1\right)& \mathrm{x}\left(\mathrm{x} - 1\right)\left(\mathrm{x} - 2\right)& \left(\mathrm{x} + 1\right)\mathrm{x}\left(\mathrm{x} - 1\right)\end{array}\right|$ then f(100) is equal to :

• 0

• 1

• 100

• - 100

If ${\mathrm{p\lambda }}^4 + {\mathrm{q\lambda }}^3 + {\mathrm{r\lambda }}^2 + \mathrm{s\lambda } + \mathrm{t} = \left|\begin{array}{ccc}{\mathrm{\lambda }}^2 + 3\mathrm{\lambda }& \mathrm{\lambda } - 1& \mathrm{\lambda } + 3\\ \mathrm{\lambda } + 1& 2 - \mathrm{\lambda }& \mathrm{\lambda } - 3\\ \mathrm{\lambda } - 3& \mathrm{\lambda } + 4& 3\mathrm{\lambda }\end{array}\right|$ then t is equal to :

• 33

• 22

• 21

• - 33