Mathematics : Matrices

In the interval $\left[- \frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{4}\right]$, the number of real solutions of the equations $\left|\begin{array}{ccc}\sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ \cos \left(\mathrm{x}\right)& \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ \cos \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \sin \left(\mathrm{x}\right)\end{array}\right|$ = 0 is

• 0

• 2

• 1

• 3

If A = $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, then A¹⁰⁰ is equal to

• 100 A

• 2⁹⁹ A

• 2¹⁰⁰ A

• 99 A

An orthogonal matrix is

• $\left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& 2\sin \left(\mathrm{\alpha }\right)\\ - 2\sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha }\right)\\ - \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha }\right)\\ \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right]$

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

Which one of the following is not true?

• Matrix addition is commutative

• Matrix addition is associative

• Matrix multiplication is commutative

• Matrix multiplication is associative

If matrix $\left[\begin{array}{ccc}0& 1& - 2\\ - 1& 0& 3\\ \mathrm{\lambda }& - 3& 0\end{array}\right]$ is singular, then $\mathrm{\lambda }$ equal to

• - 2

• - 1

• 1

• 2

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

• 2

• $- \frac{1}{2}$

• S

• $\frac{1}{2}$

If $\mathrm{a} \ne \mathrm{b} \ne \mathrm{c}$, the value of x which satisfies the equation $\left|\begin{array}{ccc}0& \mathrm{x} - \mathrm{a}& \mathrm{x} - \mathrm{b}\\ \mathrm{x} + \mathrm{a}& 0& \mathrm{x} - \mathrm{c}\\ \mathrm{x} + \mathrm{b}& \mathrm{x} + \mathrm{c}& 0\end{array}\right|$ = 0, is

• x = 0

• x = a

• x = b

• 1

For how many values of x in the closed interval [- 4, - 1], is the matrix $\left[\begin{array}{ccc}3& - 1 + \mathrm{x}& 2\\ 3& - 1& \mathrm{x} + 2\\ \mathrm{x} + 3& - 1& 2\end{array}\right]$ singular ?

• 2

• 0

• 3

• 1

$\left[\begin{array}{c}1\\ - 1\\ 2\end{array}\right]\left[\begin{array}{ccc}2& 1& - 1\end{array}\right]$ is equal to

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

• $\left[\begin{array}{ccc}2& 1& - 1\\ - 2& - 1& 1\\ 4& 2& - 2\end{array}\right]$

• [- 1]

• Not defined

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

• $\left[\begin{array}{cc}- 5& - 2\\ - 3& 1\end{array}\right]$

• $\left[\begin{array}{cc}5/11& 2/11\\ 3/11& - 1/11\end{array}\right]$

• $\left[\begin{array}{cc}- 5/11& - 2/11\\ - 3/11& - 1/11\end{array}\right]$

• $\left[\begin{array}{cc}5& 2\\ 3& - 1\end{array}\right]$