Mathematics : Matrices

If $\left|\begin{array}{ccc}- {\mathrm{a}}^2& \mathrm{ab}& \mathrm{ac}\\ \mathrm{ab}& - {\mathrm{b}}^2& \mathrm{bc}\\ \mathrm{ac}& \mathrm{bc}& - {\mathrm{c}}^2\end{array}\right|$ = ka²b²c², then k is equal to

• 2

• 4

• - 4

• 8

If $\left|\begin{array}{ccc}6\mathrm{i}& - 3\mathrm{i}& 1\\ 4& 3\mathrm{i}& - 1\\ 20& 3& \mathrm{i}\end{array}\right|$ = x + iy, then (x, y) is

• (3, 1)

• (1, 3)

• (0, 3)

• (0, 0)

If $\left[\begin{array}{ccc}1& 3& \mathrm{\lambda } + 2\\ 2& 4& 8\\ 3& 5& 10\end{array}\right]$ is a singular matrix, then the value of $\mathrm{\lambda }$ is

• - 2

• 4

• 2

• - 4

$\left|\begin{array}{ccc}1 + \mathrm{x}& 1& 1\\ 1& 1 + \mathrm{y}& 1\\ 1& 1& 1 +\hspace{0.17em}\mathrm{z}\end{array}\right|$ is equal to

• $\mathrm{xyz}\left(1 + \frac{1}{\mathrm{x}} + \frac{1}{\mathrm{y}} + \frac{1}{\mathrm{z}}\right)$

• xyz

• $1 + \frac{1}{\mathrm{x}} + \frac{1}{\mathrm{y}} + \frac{1}{\mathrm{z}}$

• $\frac{1}{\mathrm{x}} + \frac{1}{\mathrm{y}} + \frac{1}{\mathrm{z}}$

If $\left|\begin{array}{ccc}\mathrm{x}& {\mathrm{x}}^2& 1 + {\mathrm{x}}^3\\ \mathrm{y}& {\mathrm{y}}^2& 1 + {\mathrm{y}}^3\\ \mathrm{z}& {\mathrm{z}}^2& 1 + {\mathrm{z}}^3\end{array}\right| = 0$, then

• z = xy

• $\mathrm{z} = \frac{1}{\mathrm{xy}}$

• $\mathrm{z} = - \frac{1}{\mathrm{xy}}$

• None of these

A square matrix A is called an orthogonal matrix, if

• $\mathrm{A} \overline{\mathrm{A}} = \mathrm{I}$

• AA' = I

• ${\mathrm{AA}}^{\mathrm{\theta }} = \mathrm{I}$

• A² = I

If the equations 2x - y + 2z = 2, x - 2y + z = - 4 and x + y + $\mathrm{\lambda }$z = 4 have no solution, then the value of $\mathrm{\lambda }$ will be

• 1

• 2

• 3

• - 4

$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}\mathrm{\alpha }& 0\\ 1& 1\end{array}\right], \mathrm{B} = \left[\begin{array}{cc}1& 0\\ 5& 1\end{array}\right]$, then the value of $\mathrm{\alpha }$, if A² = B, will be

• 4

• 3

• 5

• None of these

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