Mathematics : Matrices

Suppose A is a matrix of order 3 and $\mathrm{B} = \left|\mathrm{A}\right|{\mathrm{A}}^{-1}$. If $\left|\mathrm{A}\right| = - 5$, then $\left|\mathrm{B}\right|$ is equal to :

• 1

• - 5

• - 1

• 25

If $\left|\begin{array}{ccc}2& - 3\mathrm{i}& 1\\ 3& 3\mathrm{i}& - 1\\ 4& 3& \mathrm{i}\end{array}\right|$ = x + iy, then :

• x = 3, y = 1

• x = 2, y = 5

• x = 0, y = 0

• x = 1, y = 1

The value of the $\left|\begin{array}{ccc}{}^{10}\mathrm{C}_4& {}^{10}\mathrm{C}_5& {}^{11}\mathrm{C}_{\mathrm{m}}\\ {}^{11}\mathrm{C}_6& {}^{11}\mathrm{C}_7& {}^{12}\mathrm{C}_{\mathrm{m} + 2}\\ {}^{12}\mathrm{C}_8& {}^{12}\mathrm{C}_9& {}^{13}\mathrm{C}_{\mathrm{m} + 4}\end{array}\right| = 0$ when m is equal to :

• 6

• 5

• 4

• 1

Let A = $\left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& - \sin \left(\mathrm{\alpha }\right)\\ \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right]$, $\mathrm{\alpha } \in \mathrm{R}$ such that A³² = $\left[\begin{array}{cc}0& - 1\\ 1& 0\end{array}\right]$ then a value of $\mathrm{\alpha }$ is

• 0

• $\frac{\mathrm{\pi }}{32}$

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{64}$

The greatest value of c $\in$ R or which the system of linear equations

x – cy – cz = 0

cx - y + cz = 0

cx + cy – z = 0 has non – trivial solution, is:

• - 1

• 0

• $\frac{1}{2}$

• 2

The total number matrices $\mathrm{A} = \left[\begin{array}{ccc}0& 2\mathrm{y}& 1\\ 2\mathrm{x}& \mathrm{y}& - 1\\ 2\mathrm{x}& - \mathrm{y}& 1\end{array}\right]$, $\left(\mathrm{x}, \mathrm{y} \in \mathrm{R}, \mathrm{x} \ne \mathrm{y}\right)$ for which A^TA = 3I₃ is

• 2

• 6

• 4

• 3

If $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right] ... \left[\begin{array}{cc}1& \mathrm{n} - 1\\ 0& 1\end{array}\right] = \left[\begin{array}{cc}1& 78\\ 0& 1\end{array}\right]$ then the inverse of $\left[\begin{array}{cc}1& \mathrm{n}\\ 0& 1\end{array}\right]$ is

• $\left[\begin{array}{cc}1& 12\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& 0\\ 12& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& 0\\ 13& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& - 13\\ 0& 1\end{array}\right]$

If the system of linear equations x + y + z = 5, x + 2y + 2z = 6, x + 3y + $\mathrm{\lambda z} = \mathrm{\mu }, (\mathrm{\lambda }, \mathrm{\mu } \in \mathrm{R})$ has infinitely many solutions, then the value of $\mathrm{\lambda } +\hspace{0.17em}\mathrm{\mu }$ is

• 7

• 10

• 12

• 9

The sum of the real roots of the equation $\left|\begin{array}{ccc}\mathrm{x}& - 6& - 1\\ 2& - 3\mathrm{x}& \mathrm{x} - 3\\ - 3& 2\mathrm{x}& \mathrm{x} + 2\end{array}\right| = 0$, is equal to

• 0

• 6

• - 4

• 1

If $\mathrm{A} = \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, then A² - 5A is equal to

• 2I

• - 2I

• 3I

• null matrix