Mathematics : Matrices

G = $\left\{\left[\begin{array}{cc}\mathrm{x}& \mathrm{x}\\ \mathrm{x}& \mathrm{x}\end{array}\right], \mathrm{x} \mathrm{is} \mathrm{a} \mathrm{non}-\mathrm{zero} \mathrm{real} \mathrm{number}\right\}$ is a roup with respect to matrix multiplication. In this group, the inverse of $\left[\begin{array}{cc}\frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{3}\end{array}\right]$ is

• $\left[\begin{array}{cc}\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.& \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\\ \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.& \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\end{array}\right]$

• $\left[\begin{array}{cc}\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.& \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\\ \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.& \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\end{array}\right]$

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

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

$\left|\begin{array}{ccc}\sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha } + \mathrm{\delta }\right)\\ \sin \left(\mathrm{\beta }\right)& \cos \left(\mathrm{\beta }\right)& \sin \left(\mathrm{\beta } + \mathrm{\delta }\right)\\ \sin \left(\mathrm{\gamma }\right)& \cos \left(\mathrm{\gamma }\right)& \sin \left(\mathrm{\gamma } + \mathrm{\delta }\right)\end{array}\right|$ is equal to

• 0

• 1

• $1 + \sin \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\beta }\right)\sin \left(\mathrm{\gamma }\right)$

• $1 - \left(\sin \left(\mathrm{\alpha }\right) - \sin \left(\mathrm{\beta }\right)\right)\left(\sin \left(\mathrm{\beta }\right) - \sin \left(\mathrm{\gamma }\right)\right)\left(\sin \left(\mathrm{\gamma }\right) - \sin \left(\mathrm{\alpha }\right)\right)$

If ax⁴ + bx³ + cx² + dx + e = $\left|\begin{array}{ccc}{\mathrm{x}}^3 + 3\mathrm{x}& \mathrm{x} - 1& \mathrm{x} +\hspace{0.17em}3\\ \mathrm{x} + 1& - 2\mathrm{x}& \mathrm{x} - 4\\ \mathrm{x} - 3& \mathrm{x} + 4& 3\mathrm{x}\end{array}\right|$, then e is equal to

• 1

• 0

• 2

• - 1

If A and B are square matnces of order n such that A² - B² = (A - B) (A + B), then which of the following will be true?

• Either A or B is zero matrix

• A = B

• AB = BA

• Either A or B is identity matrix

If A = $\left[\begin{array}{cc}\mathrm{\alpha }& 2\\ 2& \mathrm{\alpha }\end{array}\right] \mathrm{and} \left|{\mathrm{A}}^3\right| = 125$, then $\mathrm{\alpha }$ is equal to

• $\pm 1$

• $\pm 2$

• $\pm 3$

• $\pm 5$

If the matrix $\left[\begin{array}{cc}2& 3\\ 5& - 1\end{array}\right] = \mathrm{A} + \mathrm{B}$, where A is symmetric and B is skew - symmetric, then B is equal to

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

• $\left[\begin{array}{cc}0& - 2\\ 2& 0\end{array}\right]$

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

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

If A is $3 \times 4$ matrix and B is a matrix such that A'B and BA' are both defined, then B is of the type

• $4 \times 4$

• $3 \times 4$

• $4 \times 3$

• $3 \times 3$

The symmetric part of the matrix A = $\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& - 2& 7\end{array}\right]$ is

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

• $\left[\begin{array}{ccc}1& 4& 3\\ 2& 8& 0\\ 3& 0& 7\end{array}\right]$

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

• $\left[\begin{array}{ccc}1& 4& 3\\ 4& 8& 0\\ 3& 0& 7\end{array}\right]$

The inverse ofthe matrix A = $\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$ is

• $\frac{1}{24}\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$

• $\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$

• $\frac{1}{24}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

• $\left[\begin{array}{ccc}\frac{1}{2}& 0& 0\\ 0& \frac{1}{3}& 0\\ 0& 0& \frac{1}{4}\end{array}\right]$

If a, b and c are in AP, then the value of $\left|\begin{array}{ccc}\mathrm{x} + 2& \mathrm{x} + 3& \mathrm{x} + \mathrm{a}\\ \mathrm{x} + 4& \mathrm{x} + 5& \mathrm{x} +\hspace{0.17em}\mathrm{b}\\ \mathrm{x} +\hspace{0.17em}6& \mathrm{x} + 7& \mathrm{x} + \mathrm{c}\end{array}\right|$ is

• 0

• x - (a + b + c)

• a + b + c

• 9x² + a + b + c