Mathematics : Matrices

If $\left|\begin{array}{cc}1 - \mathrm{i}& \mathrm{i}\\ 1 + 2\mathrm{i}& - \mathrm{i}\end{array}\right| = \mathrm{x} +\hspace{0.17em}\mathrm{iy}$, then x is equal to

• - 2

• - 1

• 1

• 2

If A, B are square matrices of order 3. A is nonsingular and AB = 0, then B is

• null matrix

• non singular matrix

• singular matrix

• unit matrix

$$\begin{gathered} \mathrm{If} \mathrm{a}\ne \mathrm{p}, \mathrm{b} \ne \mathrm{q}, \mathrm{c} \ne 0 \mathrm{and} \\ \left|\begin{array}{ccc}\mathrm{p}& \mathrm{b}& \mathrm{c}\\ \mathrm{p} + \mathrm{a}& \mathrm{q} + \mathrm{b}& 2\mathrm{c}\\ \mathrm{a}& \mathrm{b}& \mathrm{r}\end{array}\right| = 0, \mathrm{then} \\ \frac{\mathrm{p}}{\mathrm{p} - \mathrm{a}} + \frac{\mathrm{q}}{\mathrm{q} - \mathrm{b}} + \frac{\mathrm{r}}{\mathrm{r} - \mathrm{c}} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• 0

• 1

• 2

• 3

The number of solutions of the system of equations 2x + y - z = 7, x - 3y + 2z = 1, x + 4y - 3z = 5 is

• 0

• 1

• 2

• 3

$$\begin{gathered} \mathrm{If} \left|\begin{array}{ccc}\cos \left(\mathrm{A} + \mathrm{B}\right)& - \sin \left(\mathrm{A} + \mathrm{B}\right)& \cos \left(2\mathrm{B}\right)\\ \sin \left(\mathrm{A}\right)& \cos \left(\mathrm{A}\right)& \sin \left(\mathrm{B}\right)\\ - \cos \left(\mathrm{A}\right)& \sin \left(\mathrm{A}\right)& \cos \left(\mathrm{B}\right)\end{array}\right| = 0 \\ \mathrm{then} \mathrm{B} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\left(2\mathrm{n} + 1\right)\frac{\mathrm{\pi }}{2}$

• $\left(2\mathrm{n} + 1\right)$

• $\mathrm{n\pi }$

• $2\mathrm{n\pi }$

The value of $\left|\begin{array}{ccc}1990& 1991& 1992\\ 1991& 1992& 1993\\ 1992& 1993& 1994\end{array}\right|$ is

• 1992

• 1993

• 1994

• 0

The rank of $\left[\begin{array}{ccc}1& - 1& 1\\ 1& 1& - 1\\ - 1& 1& 1\end{array}\right]$ is

• 0

• 1

• 2

• 3

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

• 2A

• 2I

• A

• I

$\mathrm{A} = \left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right] \mathrm{then} {\mathrm{A}}^3 - 4{\mathrm{A}}^2 - 6\mathrm{A} \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 0

• A

• - A

• I

$\left|\begin{array}{ccc}\log \left(\mathrm{e}\right)& \log \left({\mathrm{e}}^2\right)& \log \left({\mathrm{e}}^3\right)\\ \log \left({\mathrm{e}}^2\right)& \log \left({\mathrm{e}}^3\right)& \log \left({\mathrm{e}}^4\right)\\ \log \left({\mathrm{e}}^3\right)& \log \left({\mathrm{e}}^4\right)& \log \left({\mathrm{e}}^5\right)\end{array}\right| \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 0

• 1

• 4log(e)

• 5log(e)