Mathematics : Matrices

If  = x + iy, then

• x = 3, y = 1

• x = 1, y = 3

• x = 0, y = 3

• x = 0, y = 0

 is equal to

• 1

• 0

• ${\log }_{\mathrm{a}}\left(\mathrm{b}\right)$

• ${\log }_{\mathrm{b}}\left(\mathrm{a}\right)$

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

• A² = A

• B² = B

• $\mathrm{AB} \ne \mathrm{BA}$

• AB = BA

If A = $\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{b}& \mathrm{a}\end{array}\right] \mathrm{and} {\mathrm{A}}^2 = \left[\begin{array}{cc}\mathrm{\alpha }& \mathrm{\beta }\\ \mathrm{\beta }& \mathrm{\alpha }\end{array}\right]$, then

• $\mathrm{\alpha } = {\mathrm{a}}^2 + {\mathrm{b}}^2, \mathrm{\beta } = \mathrm{ab}$

• $\mathrm{\alpha } = {\mathrm{a}}^2 + {\mathrm{b}}^2, \mathrm{\beta } = 2\mathrm{ab}$

• $\mathrm{\alpha } = {\mathrm{a}}^2 + {\mathrm{b}}^2, \mathrm{\beta } = {\mathrm{a}}^2 - {\mathrm{b}}^2$

• $\mathrm{\alpha } = 2\mathrm{ab}, \mathrm{\beta }= {\mathrm{a}}^2 + {\mathrm{b}}^2$

If matrix A = $\left[\begin{array}{cc}1& - 1\\ 1& 1\end{array}\right]$, then

• $\mathrm{A}\text{'} = \left[\begin{array}{cc}1& 1\\ 1& - 1\end{array}\right]$

• ${\mathrm{A}}^{- 1} = \left[\begin{array}{cc}1& 1\\ - 1& 1\end{array}\right]$

• $\mathrm{A} . \left[\begin{array}{cc}1& 1\\ - 1& 1\end{array}\right] = 2\mathrm{I}$

• $\mathrm{\lambda A} = \left[\begin{array}{cc}\mathrm{\lambda }& - \mathrm{\lambda }\\ - 1& 1\end{array}\right], \mathrm{where} \mathrm{\lambda } \mathrm{is} \mathrm{a} \mathrm{non}- \mathrm{zero} \mathrm{scalar}$

If for AX = B, B = $\left[\begin{array}{c}9\\ 52\\ 0\end{array}\right] \mathrm{and} {\mathrm{A}}^{- 1} = \left[\begin{array}{ccc}3& - 1/2& - 1/2\\ - 4& 3/4& 5/4\\ 2& - 1/4& - 3/4\end{array}\right]$, then X is equal to

• $\left[\begin{array}{c}1\\ 3\\ 5\end{array}\right]$

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

• $\left[\begin{array}{c}- 4\\ 2\\ 3\end{array}\right]$

• $\left[\begin{array}{c}3\\ 3/4\\ - 3/4\end{array}\right]$

The value of $\left|\begin{array}{ccc}\mathrm{y} +\hspace{0.17em}\mathrm{z}& \mathrm{x}& \mathrm{x}\\ \mathrm{y}& \mathrm{z} + \mathrm{x}& \mathrm{y}\\ \mathrm{z}& \mathrm{z}& \mathrm{x} +\hspace{0.17em}\mathrm{y}\end{array}\right|$ is

• xyz

• x²y²z²

• 4xyz

• 4x²y²z²

If A and B are square matrices of order 3 such that $\left|\mathrm{A}\right|$ = - 1, $\left|\mathrm{B}\right|$ = 3, then $\left|3\mathrm{AB}\right|$ is equal to

• - 9

• - 81

• - 27

• 81

Let A = $\left[\begin{array}{ccc}0& 0& - 1\\ 0& - 1& 0\\ - 1& 0& 0\end{array}\right]$, the only correct statement about the matrix A, is

• A is a zero matrix

• A = (- 1)I, where I is a unit matrix

• A^{- 1} does not exist

• A² = I

If A = $\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$, then A² is equal to

• 5A

• 10A

• 16A

• 32A