Determinants

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

• $- \frac{1}{2}\left[\begin{array}{cc}4& - 2\\ - 3& 1\end{array}\right]$

• $\frac{1}{2}\left[\begin{array}{cc}4& - 2\\ - 3& 1\end{array}\right]$

• $\left[\begin{array}{cc}- 2& 4\\ 1& 3\end{array}\right]$

• $\left[\begin{array}{cc}2& 4\\ 1& 3\end{array}\right]$

If A = $\left[\begin{array}{ccc}2& 1& 0\\ 0& 2& 1\\ 1& 0& 2\end{array}\right]$, then $\left|\mathrm{adj}\right|$ is equal to

• 0

• 9

• $\frac{1}{9}$

• 81

If f(x) = $\left|\begin{array}{ccc}\sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \tan \left(\mathrm{x}\right)\\ {\mathrm{x}}^3& {\mathrm{x}}^2& \mathrm{x}\\ 2\mathrm{x}& 1& \mathrm{x}\end{array}\right|$, then $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)}{{\mathrm{x}}^2}$ is equal to

• 0

• 3

• 2

• 1

Inverse of a diagonal non-singular matrix is

• scalar matrix

• skew symmetric matrix

• zero matrix

• diagonal matrix

The characteristic equation of a matrix A is  ${\mathrm{\lambda }}^3 - 5{\mathrm{\lambda }}^2 - 3\mathrm{\lambda } + 2 = 0$, then $\left|\mathrm{adj} \left(\mathrm{A}\right)\right|$ is equal to

• 9

• 25

• $\frac{1}{2}$

• 4

If $\mathrm{A} = \left|\begin{array}{ccc}\mathrm{x}& 1& 1\\ 1& \mathrm{x}& 1\\ 1& 1& \mathrm{x}\end{array}\right| \mathrm{and} \mathrm{B} = \left|\begin{array}{cc}\mathrm{x}& 1\\ 1& \mathrm{x}\end{array}\right|$, then $\frac{\mathrm{dA}}{\mathrm{dx}}$ is equal to

• 3B + 1

• 3B

• - 3B

• 1 - 3B

If the determinant of the adjoint of a (real) matrix of order 3 is 25, then the determinant of the inverse of the matrix is

• 0.2

• $\pm 5$

• $\frac{1}{\sqrt[5]{625}}$

• $\pm 0.2$

If A is a matrix of order 3 such that A (adj A) = 10I, then $\left|\mathrm{adj} \mathrm{A}\right|$ is equal to

• 1

• 10

• 100

• 10I

Consider the following statements

(i) If any two rows or columns of a determinant are identical, then the value of the determinant is zero.
(ii) Ifthe corresponding rows and columns of a determinant are interchanged, then the value of determmant does not change.
(iii) If any two rows (or columns) of a determinant are mterchanged, then the value of the determinant changes in sign.

Which of these are correct?

• (i) and (iii)

• (i) and (ii)

• (i), (ii) and (iii)

• (ii) and (iii)

If P = $\left|\begin{array}{cc}\mathrm{x}& 1\\ 1& \mathrm{x}\end{array}\right|$ and Q = $\left|\begin{array}{ccc}\mathrm{x}& 1& 1\\ 1& \mathrm{x}& 1\\ 1& 1& \mathrm{x}\end{array}\right|$, then $\frac{\mathrm{dQ}}{\mathrm{dX}}$ is equal to

• - 3P

• 3P + 1

• 3P

• 1 - 3P