What are the roots of the equation

$\left|{\mathrm{x}}^2 - \mathrm{x} - 6\right| = \mathrm{x} + 2$ ?

•  - 2, 1, 4

• 0, 2, 4

• 0, 1, 4

•  - 2, 2, 4

$\mathrm{If} \left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right], \mathrm{then} \mathrm{the} \mathrm{matrix} \mathrm{A} \mathrm{is} \mathrm{a}/\mathrm{an} :$

• Singular matrix

• Involutory matrix

• Nilpotent matrix

• Idempotent matrix

$\mathrm{If} \left[\begin{array}{ccc}\mathrm{x}& - 3\mathrm{i}& 1\\ \mathrm{y}& 1& \mathrm{i}\\ 0& 2\mathrm{i}& - \mathrm{i}\end{array}\right] = 6 + 11\mathrm{i}, \mathrm{then} \mathrm{what} \mathrm{are} \mathrm{the} \mathrm{values} \mathrm{of} \mathrm{x} \mathrm{and} \mathrm{y} \mathrm{respectively} ?$

•  - 3, 4

• 3, 4

• 3, - 4

•  - 3,  - 4

If A is a square matrix of order n > 1, then which one of the following is correct ?

• det ( - A) = det A

• det( - A) = ( - 1)ⁿ det(A)

• det( - A) = - det(A)

• det( - A) = ndet(A)

What is the inverse of the matrix

$\mathrm{A} = \left(\begin{array}{ccc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)& 0\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)& 0\\ 0& 0& 1\end{array}\right) = ?$

• $\left(\begin{array}{ccc}\cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)& 0\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)& 0\\ 0& 0& 1\end{array}\right)$

• $\left(\begin{array}{ccc}\cos \left(\mathrm{\theta }\right)& 0& - \sin \left(\mathrm{\theta }\right)\\ 0 & 1& 0\\ \sin \left(\mathrm{\theta }\right)& 0& \cos \left(\mathrm{\theta }\right)\end{array}\right)$

• $\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ 0& \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right)$

• $\left(\begin{array}{ccc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)& 0\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)& 0\\ 0& 0& 1\end{array}\right)$

If A is a 2 x 3 matrix and AB is a 2 x 5 matrix, then B must be a : 

• 3 x 5 Matrix

• 5 x 3 Matrix

• 3 x 2 Matrix

• 5 x 2 Matrix

$\mathrm{If} \mathrm{A} = \left(\begin{array}{cc}1& 2\\ 2& 3\end{array}\right) \mathrm{and} {\mathrm{A}}^2 - \mathrm{kA} - {\mathrm{I}}_2 = 0, \mathrm{where} {\mathrm{I}}_2 \mathrm{is} \mathrm{the} 2 \mathrm{x} 2 \mathrm{identity} \mathrm{matrix}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{k} ?$

• 4

•  - 4

• 8

•  - 8

8.

A square matrix A is called orthogonal if

• A = A²

• $\mathrm{A}\text{'} = {\mathrm{A}}^{ - 1}$

• $\mathrm{A} = {\mathrm{A}}^{ - 1}$

• $\mathrm{A} = \mathrm{A}\text{'}$

What is the adjoint of the matrix

$\left(\begin{array}{cc}\cos \left( - \mathrm{\theta }\right)& - \sin \left( - \mathrm{\theta }\right)\\ - \sin \left( - \mathrm{\theta }\right)& \cos \left( - \mathrm{\theta }\right)\end{array}\right) ?$

• $\left(\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right) ?$

• $\left(\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right)$

• $\left(\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right)$

• $\left(\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right)$

For a square matrix A, which of the following properties hold?

${\left({\mathrm{A}}^{ - 1}\right)}^{ - 1} = \mathrm{A}$

$$\begin{gathered} 2. \det \left({\mathrm{A}}^{ - 1}\right) = \frac{1}{\det \mathrm{A}} \\ 3. {\left(\mathrm{\lambda A}\right)}^{ - 1} = {\mathrm{\lambda A}}^{ - 1} \mathrm{where} \mathrm{\lambda } \mathrm{is} \mathrm{a} \mathrm{scalar} \\ \mathrm{Select} \mathrm{the} \mathrm{correct} \mathrm{answer} \mathrm{using} \mathrm{the} \mathrm{code} \mathrm{given} \mathrm{below} : \end{gathered}$$

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3