Mathematics : Matrices

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

• A + B = B + A and A + (B + C) = (A + B) + C

• A + B = B + A and AC = BC

• A + B = B + A and AB = BC

• AC = BC and A = BC

$\mathrm{A} = \left[\begin{array}{cc}- 2& 4\\ - 1& 2\end{array}\right]$, then A² is equal to

• null matrix

• unit matrix

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

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

If A = [x y z], B = $\left[\begin{array}{ccc}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{h}& \mathrm{b}& \mathrm{f}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}\end{array}\right] \mathrm{and} \mathrm{C} = \left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]$. Then, ABC = O, if

• [ax² + by² + cz² + 2gxy + 2fyz + 2czx] = 0

• [ax² + cy² + bz² + xy + yz + zx] = 0

• [ax² + by² + cz² + 2hxy + 2by + 2cz] = 0

• [ax² + by² + cz² + 2zx + 2hxy + 2fyz] = 0

A = $\left[\begin{array}{ccc}0& 3& 3\\ - 3& 0& - 4\\ - 3& 4& 0\end{array}\right] \mathrm{and} \mathrm{B} = \left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]$, then B'(AB) is

• null matrix

• singular matrix

• unit matrix

• symmetric matrix

A square matrix is an orthogonal matrix, if

• AA' = 0

• A + A' = I

• AA' = I

• None of these

$\mathrm{A} = \left[\begin{array}{cc}1& 2\\ 3& 2\\ - 1& 0\end{array}\right], \mathrm{B} = \left[\begin{array}{ccc}1& 3& 2\\ 4& - 1& 3\end{array}\right]$, then order of AB is

• 2 x 2

• 3 x 3

• 1 x 3

• 3 x 2

If A + I = $\left[\begin{array}{cc}3& - 2\\ 4& 1\end{array}\right]$, then (A + I)(A - I) is equal to

• $\left[\begin{array}{cc}- 5& - 4\\ 8& - 9\end{array}\right]$

• $\left[\begin{array}{cc}- 5& 4\\ - 8& 9\end{array}\right]$

• $\left[\begin{array}{cc}5& 4\\ 8& 9\end{array}\right]$

• $\left[\begin{array}{cc}- 5& - 4\\ - 8& - 9\end{array}\right]$

If A = $\left[\begin{array}{cc}1& - 1\\ 2& - 1\end{array}\right]$ and B = $\left[\begin{array}{cc}1& \mathrm{a}\\ 4& \mathrm{b}\end{array}\right]$ and (A + B)² = A² + B². Then, a and b are respectively

• 1, - 1

• 2, - 3

• - 1, 1

• 3, - 2

If A = $\left[\begin{array}{ccc}2& - 1& 1\\ - 2& 3& - 2\\ - 4& 4& - 3\end{array}\right]$, then A² is equal to

• null matrix

• itself A

• unit matrix

• Scalar matrix

If $\mathrm{A}\left(\mathrm{\alpha }\right) = \left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha }\right)\\ - \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right]$, then the matrix ${\mathrm{A}}^2\left(\mathrm{\alpha }\right)$ is

• $\mathrm{A}\left(2\mathrm{\alpha }\right)$

• $\mathrm{A}\left(\mathrm{\alpha }\right)$

   

• $\mathrm{A}\left(3\mathrm{\alpha }\right)$

• $\mathrm{A}\left(4\mathrm{\alpha }\right)$