If a, b, c and d are real numbers such that ${\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2 + {\mathrm{d}}^2 = 1 \mathrm{and} \mathrm{A} = \left[\begin{array}{cc}\mathrm{a} + \mathrm{id}& \mathrm{c} + \mathrm{id}\\ - \mathrm{c} + \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right], \mathrm{then} {\mathrm{A}}^{ - 1} = ?$

• $\left[\begin{array}{cc}\mathrm{a} +ib& - \mathrm{c} - \mathrm{id}\\ - \mathrm{c} - \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right]$

• $\left[\begin{array}{cc}\mathrm{a} - \mathrm{ib}& \mathrm{c} - \mathrm{id}\\ - \mathrm{c} + \mathrm{id}& \mathrm{a} + \mathrm{ib}\end{array}\right]$

• $\left[\begin{array}{cc}\mathrm{a} - \mathrm{ib}& - \mathrm{c} - \mathrm{id}\\ \mathrm{c} - \mathrm{id}& \mathrm{a} + \mathrm{ib}\end{array}\right]$

• $\left[\begin{array}{cc}\mathrm{a} +\mathrm{id}& \mathrm{c} + \mathrm{id}\\ \mathrm{c} - \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right]$

$\mathrm{If} \mathrm{the} \mathrm{matrix} \mathrm{A} = \left[\begin{array}{cccc}1& 2& 3& 0\\ 2& 4& 3& 2\\ 3& 2& 1& 3\\ 6& 8& 7& \mathrm{\alpha }\end{array}\right] \mathrm{is} \mathrm{of} \mathrm{rank} 3, \mathrm{then} \mathrm{\alpha } = ?$

• - 5

• 5

• 4

• 1

$\mathrm{If} \mathrm{k} > 1 \mathrm{and} \mathrm{the} \mathrm{determinant} \mathrm{of} \mathrm{the} \mathrm{matrix} {\mathrm{A}}^2, \mathrm{where} \mathrm{A} = \left[\begin{array}{ccc}\mathrm{k}& \mathrm{k\alpha }& \mathrm{\alpha }\\ 0& \mathrm{\alpha }& \mathrm{k\alpha }\\ 0& 0& \mathrm{k}\end{array}\right], \mathrm{is} {\mathrm{k}}^2, \mathrm{then} \left|\mathrm{\alpha }\right| = ?$

• $\frac{1}{{\mathrm{k}}^2}$

• k

• k²

• $\frac{1}{\mathrm{k}}$

524.

If A is a square matrix of order 3, then $\left|\mathrm{adj} (\mathrm{adj} {\mathrm{A}}^2)\right|$ is equal to

• ${\left|\mathrm{A}\right|}^2$

• ${\left|\mathrm{A}\right|}^4$

• ${\left|\mathrm{A}\right|}^8$

• ${\left|\mathrm{A}\right|}^{16}$

The system 2x + 3y + z = 5, 3x + y + 5z = 7 and x + 4y - 2z = 3 has

• unique solution

• finite number of solution

• Infinite solutions

• No solution

If $∆ = \left|\begin{array}{ccc}1& \cos \left(\mathrm{\theta }\right)& 1\\ - \cos \left(\mathrm{\theta }\right)& 1& \cos \left(\mathrm{\theta }\right)\\ - 1& - \cos \left(\mathrm{\theta }\right)& 1\end{array}\right|$, then $∆$ lies in the interval

• [2, 4]

• (2, 4)

• [1, 4]

• [- 1, 1]

If a, b, c are non-zero real numbers and if the equations (a - 1) x = y + z, (b - 1)y = z + x, (c - 1)z = x + y have a non-trivial solution, then ab + be + ca = ?

• a²b²c²

• 0

• abc

• a + b + c

If a system of three linear equations in three unknowns, which is in the matrix equation form of AX = D, is in consistent, then rank of A/rank of AD is

• Less than one

• Greater than or equal to one

• One

• Greater than one

Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| $\ne$ 0. Consider the following two statements;

(P)If A $\ne$ I₂, then |A| = – 1

(Q)If |A| = 1, then tr(A) = 2

where I₂ denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then :

• (P) is true and (Q) are false

• Both (P) and (Q) are true

• Both (P) and (Q) are false

• (P) is false and (Q) is true

Let A = {x = (x, y, z)^T : PX = 0 and x² + y² + z² = 1}, where $\left[\begin{array}{ccc}1& 2& 1\\ - 2& 3& - 4\\ 1& 9& - 1\end{array}\right]$ P then the set A

• is a singleton

• contains more than two elements

• contains exactly two elements

• is an empty set.