If A and B are two invertible square matrices of same· order, then what is (AB) ^{- 1} ?

• ${\mathrm{B}}^{ - 1}{\mathrm{A}}^{ - 1}$

• ${\mathrm{A}}^{ - 1}{\mathrm{B}}^{ - 1}$

• ${\mathrm{B}}^{ - 1}\mathrm{A}$

• ${\mathrm{A}}^{ - 1}\mathrm{B}$

if a + b + c = 0, then one of the solutions of

$\left|\begin{array}{ccc}\mathrm{a} - \mathrm{x}& \mathrm{c}& \mathrm{b}\\ \mathrm{c}& \mathrm{b} - \mathrm{x}& \mathrm{a}\\ \mathrm{b}& \mathrm{a}& \mathrm{c} - \mathrm{a}\end{array}\right| = 0 \mathrm{is}$

• x = a

• $\mathrm{x} = \sqrt{\frac{3\left({\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2\right)}{2}}$

• $\mathrm{x} = \sqrt{\frac{2\left({\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2\right)}{3}}$

• x = 0

What should be the value of x so that the matrix

$\left(\begin{array}{cc}2& 4\\ - 8& \mathrm{x}\end{array}\right)$ does not have an inverse ?

• 16

•  - 16

• 8

•  - 8

Consider the following in respect of matrices A. B and C of same order :

$$\begin{gathered} 1. \left(\mathrm{A} + \mathrm{B} +\mathrm{C}\right)\text{'} = \mathrm{A}\text{'} + \mathrm{B}\text{'} + \mathrm{C}\text{'} \\ 2. \left(\mathrm{AB}\right)\text{'} = \mathrm{A}\text{'}\mathrm{B}\text{'} \\ 3. \left(\mathrm{ABC}\right)\text{'} = \mathrm{C}\text{'}\mathrm{B}\text{'}\mathrm{A}\text{'} \end{gathered}$$

Where A' is the transpose of the matrix A.

Which of the above are correct ?

• 1and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

15.

Let matrix B be the adjoint of a square matrix A, l be the identity matrix of same order as A. If k($\ne 0$) is the determinant of the matrix A, then what is AB equal to ?

• I

• kI

• k²I

• $\left(\frac{1}{\mathrm{k}}\right)\mathrm{I}$

Consider the following in respect of matrices A and B of same order :

1. A² - B² = (A + B)(A - B)
2. (A - I)(I + A) = O $\iff$ A² = I                            where I is the identity matrix and O is the null matrix.Which of the above is/are correct ?

• 1 only

• 2 only

• both 1 and 2

• neither 1 nor 2

If B is a non-singular matrix and A is a square matrix, then the value of det (B ^{- 1} AB) is equal to

• det(B)

• det(A)

• $\det \left({\mathrm{B}}^{ - 1}\right)$

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

$$\begin{gathered} \mathrm{If} \mathrm{a} \ne \mathrm{b} \ne \mathrm{c}, \mathrm{then} \mathrm{one} \mathrm{value} \mathrm{of} \mathrm{x} \mathrm{which} \mathrm{satisfies} \mathrm{the} \mathrm{equation} \\ \left|\begin{array}{ccc}0& \mathrm{x} - \mathrm{a}& \mathrm{x} - \mathrm{b}\\ \mathrm{x} + \mathrm{a}& 0& \mathrm{x} - \mathrm{c}\\ \mathrm{x} +\hspace{0.17em}\mathrm{b}& \mathrm{x} + \mathrm{c}& 0\end{array}\right| = 0 \mathrm{is} \mathrm{given} \mathrm{by} \end{gathered}$$

• a

• b

• c

• 0

$$\begin{gathered} \mathrm{If} \mathrm{A} = \left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha }\right)\\ - \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right] \mathrm{then} \mathrm{what} \mathrm{is} {\mathrm{AA}}^{\mathrm{T}} = ? \\ \left(\mathrm{Where} {\mathrm{A}}^{\mathrm{T}} \mathrm{is} \mathrm{the} \mathrm{transpose} \mathrm{of} \mathrm{A}\right) ? \end{gathered}$$

• Null matrix 

• Identity Matrix

• A

• - A

$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$, then what is A³ = ?

• $\left[\begin{array}{cc}\cos \left(3\mathrm{\theta }\right)& \sin \left(3\mathrm{\theta }\right)\\ - \sin \left(3\mathrm{\theta }\right)& \cos \left(3\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}{\cos }^3\left(\mathrm{\theta }\right)& {\sin }^3\left(\mathrm{\theta }\right)\\ - {\sin }^3\left(\mathrm{\theta }\right)& {\cos }^3\left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(3\mathrm{\theta }\right)& - \sin \left(3\mathrm{\theta }\right)\\ \sin \left(3\mathrm{\theta }\right)& \cos \left(3\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}{\cos }^3\left(\mathrm{\theta }\right)& - {\sin }^3\left(\mathrm{\theta }\right)\\ {\sin }^3\left(\mathrm{\theta }\right)& {\cos }^3\left(\mathrm{\theta }\right)\end{array}\right]$