Determinants

Value of the determinant $\left|\begin{array}{ccc}1 +\hspace{0.17em}\mathrm{a}& 1\mathrm{a}& 1\\ 1& 1 + \mathrm{b}& 1\\ 1& 1& 1 +\hspace{0.17em}\mathrm{c}\end{array}\right|$ is :

• 1 + abc + ab + bc + ca

• abc

• 4abc

• $\mathrm{abc}\left(1 + \frac{1}{\mathrm{a}} + \frac{1}{\mathrm{b}} + \frac{1}{\mathrm{c}}\right)$

If the value of the determinant $\left|\begin{array}{ccc}\mathrm{x} + 1& 1& 1\\ 2& \mathrm{x} + 2& 2\\ 3& 3& \mathrm{x} + 3\end{array}\right|$ is equal to zero, then x is :

• 0 and - 6

• 0 and 6

• 6

• - 6

If I is the unit matrix of order 10, then the determinant of I is equal to :

• 10

• 1

• $\frac{1}{10}$

• 9

$\left|\begin{array}{ccc}\mathrm{x} - 2& 2\mathrm{x} - 3& 3\mathrm{x} - 4\\ \mathrm{x} - 4& 2\mathrm{x} - 9& 3\mathrm{x} - 16\\ \mathrm{x} - 8& 2\mathrm{x} - 27& 3\mathrm{x} - 64\end{array}\right|$ = 0, then x is equal to :

• - 2

• 3

• - 4

• 4

Let A be a square matrix of order 3. If $\left|\mathrm{A}\right|$ = - 2, then the value of determinant of $\left|\mathrm{A}\right|$ adj A is :

• 8

• - 8

• - 1

• 32

If f(x) = $\left|\begin{array}{ccc}\mathrm{x} - 3& 2{\mathrm{x}}^2 - 18& 3{\mathrm{x}}^3 - 81\\ \mathrm{x} - 5& 2{\mathrm{x}}^2 - 50& 4{\mathrm{x}}^3 - 500\\ 1& 2& 3\end{array}\right|$, then f(1) . f(3) + f(3) . f(5) + f(5) . f(1) is equal to

• f(1)

• f(3)

• f(1) + f(3)

• f(1) + f(5)

If A = $\left[\begin{array}{ccc}1& - 1& 1\\ 0& 2& - 3\\ 2& 1& 0\end{array}\right]$ and B = (adj A), and C = 5A, then $\frac{\left|\mathrm{adj} \mathrm{B}\right|}{\left|\mathrm{C}\right|}$ is equal to :

• 5

• 25

• - 1

• 1

If A_i = $\left[\begin{array}{cc}{\mathrm{a}}^{\mathrm{i}}& {\mathrm{b}}^{\mathrm{i}}\\ {\mathrm{b}}^{\mathrm{i}}& {\mathrm{a}}^{\mathrm{i}}\end{array}\right]$ and, if $\left|\mathrm{a}\right| < 1, \left|\mathrm{b}\right| < 1$, then $\sum _{\mathrm{i} = 1}^{\infty }\det \left({\mathrm{A}}_{\mathrm{i}}\right)$ is equa to :

• $\frac{{\mathrm{a}}^2}{\left(1 - {\mathrm{a}}^2\right)} - \frac{{\mathrm{b}}^2}{\left(1 - {\mathrm{b}}^2\right)}$

• $\frac{{\mathrm{a}}^2 - {\mathrm{b}}^2}{\left(1 - {\mathrm{a}}^2\right)\left(1 - {\mathrm{b}}^2\right)}$

• $\frac{{\mathrm{a}}^2}{\left(1 - {\mathrm{a}}^2\right)} + \frac{{\mathrm{b}}^2}{\left(1 - {\mathrm{b}}^2\right)}$

• $\frac{{\mathrm{a}}^2}{\left(1 + {\mathrm{a}}^2\right)} - \frac{{\mathrm{b}}^2}{\left(1 + {\mathrm{b}}^2\right)}$

The value of determinant, $\left|\begin{array}{ccc}\sqrt{13} + \sqrt{3}& 2\sqrt{5}& \sqrt{5}\\ \sqrt{15} + \sqrt{26}& 5& \sqrt{10}\\ 3 + \sqrt{65}& \sqrt{15}& 5\end{array}\right|$ is :

• $5\left(\sqrt{6} - 5\right)$

• $5\sqrt{3}\left(\sqrt{6} - 5\right)$

• $\sqrt{5}\left(\sqrt{6} - \sqrt{3}\right)$

• $\sqrt{2}\left(\sqrt{7} - \sqrt{5}\right)$

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

• $\mathrm{abc}\left(1 + \frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} + \frac{\mathrm{z}}{\mathrm{c}}\right)$

• $\mathrm{abc}\left(1 + \frac{\mathrm{a}}{\mathrm{x}} + \frac{\mathrm{b}}{\mathrm{y}} + \frac{\mathrm{c}}{\mathrm{z}}\right)$

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

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