Matrices

If A = $\left[\begin{array}{ccc}2& 3& 1\\ 0& - 1& 5\end{array}\right]$ and B = $\left[\begin{array}{ccc}1& 2& - 6\\ 0& - 1& 3\end{array}\right]$, then 3A - 4B is equal to

• $\left[\begin{array}{ccc}2& 1& 27\\ 0& 1& 3\end{array}\right]$

• $\left[\begin{array}{ccc}2& 1& 27\\ 0& 1& 3\end{array}\right]$

• $\left[\begin{array}{ccc}3& 5& - 5\\ 0& - 2& 8\end{array}\right]$

• None of these

If x is a complex root of the equation $\left|\begin{array}{ccc}1& \mathrm{x}& \mathrm{x}\\ \mathrm{x}& 1& \mathrm{x}\\ \mathrm{x}& \mathrm{x}& 1\end{array}\right| + \left|\begin{array}{ccc}1 - \mathrm{x}& 1& 1\\ 1& 1 - \mathrm{x}& 1\\ 1& 1& 1 - \mathrm{x}\end{array}\right| = 0$, then ${\mathrm{x}}^{2005} + \frac{1}{{\mathrm{x}}^{2005}}$ is equal to

• 1

• - 1

• i

• w

If A² - A + I = 0, then matrix A^-1 will be equal to

• A + I

• I - A

• A

• A - I

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

• 2

• 1

• 0

• 3

If A = $\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 1& 0& 0\end{array}\right]$, then $\mathrm{A}$ is a

• singular

• non-singular

• symmetric

• unit matrix

Let $\mathrm{f}\left(\mathrm{x}\right) = \left|\begin{array}{ccc}\sin \left(3\mathrm{x}\right)& 1& 2{\left(\cos \left(\frac{3\mathrm{x}}{2}\right) + \sin \left(\frac{3\mathrm{x}}{2}\right)\right)}^2\\ \cos \left(3\mathrm{x}\right)& - 1& 2{\cos }^2\left(\frac{3\mathrm{x}}{2}\right) - {\sin }^2\left(\frac{3\mathrm{x}}{2}\right)\\ \tan \left(3\mathrm{x}\right)& 4& 1 + 2\tan \left(3\mathrm{x}\right)\end{array}\right|$. Then, the value of f'(x) at x = (2n + 1) $\mathrm{\pi }$, n $\in \mathrm{I}$ (the set of integers) is equal to

• (- 1)ⁿ

• (- 1)^{n + 1}

• 3

• 9

The value of $\left|\begin{array}{ccc}1& \mathrm{a}& \mathrm{b} + \mathrm{c}\\ 1& \mathrm{b}& \mathrm{c}\hspace{0.17em}+ \mathrm{a}\\ 1& \mathrm{c}& \mathrm{a} +\hspace{0.17em}\mathrm{b}\end{array}\right|$ is

• 0

• a + b + c

• abc

• 1

If 2x - $\left[\begin{array}{cc}1& 2\\ 7& 4\end{array}\right] = \left[\begin{array}{cc}3& 2\\ 0& - 2\end{array}\right]$, then the value of x will be

• $\left[\begin{array}{cc}2& 2\\ 7& 4\end{array}\right]$

• $\left[\begin{array}{cc}1& 2\\ 7/2& 2\end{array}\right]$

• $\left[\begin{array}{cc}2& 2\\ 7/2& 1\end{array}\right]$

• None of the above

If $\left[\begin{array}{ccc}0& 2\mathrm{\beta }& \mathrm{\gamma }\\ \mathrm{\alpha }& \mathrm{\beta }& - \mathrm{\gamma }\\ \mathrm{\alpha }& - \mathrm{\beta }& \mathrm{\gamma }\end{array}\right]$ is orthagonal, then the values of $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ will be

• $\mathrm{\alpha } = \pm \frac{1}{\sqrt{5}}, \mathrm{\beta } = \pm \frac{1}{\sqrt{3}}, \mathrm{\gamma } = \pm \frac{1}{\sqrt{2}}$

• $\mathrm{\alpha } = \pm \frac{1}{\sqrt{6}}, \mathrm{\beta } = \pm \frac{1}{\sqrt{7}}, \mathrm{\gamma } = \pm \frac{1}{\sqrt{3}}$

• $\mathrm{\alpha } = \pm \frac{1}{\sqrt{3}}, \mathrm{\beta } = \pm \frac{1}{\sqrt{3}}, \mathrm{\gamma } = \pm \frac{1}{\sqrt{3}}$

• $\mathrm{\alpha } = \pm \frac{1}{\sqrt{2}}, \mathrm{\beta } = \pm \frac{1}{\sqrt{6}}, \mathrm{\gamma } = \pm \frac{1}{\sqrt{3}}$

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

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

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

• $\frac{1}{10}\left[\begin{array}{cc}2& 2\\ 2& 3\end{array}\right]$

• $\frac{1}{10}\left[\begin{array}{cc}3& 2\\ - 2& 2\end{array}\right]$