Determinants

If A = $\left[\begin{array}{cc}\mathrm{a}& \mathrm{x}\\ \mathrm{y}& \mathrm{a}\end{array}\right]$ and if xy = 1, then det (AA^T) is equal to

• a² - 1

• (a² + 1)²

• 1 - a²

• (a² - 1)²

If A = $\left[\begin{array}{cc}\mathrm{x}& \mathrm{x} - 1\\ 2\mathrm{x}& 1\end{array}\right]$ and if det A = - 9, then the avlue of x are

• $\frac{3}{2}, - 3$

• $\frac{- 2}{3}, 3$

• $\frac{2}{3}, 3$

• $- \frac{3}{2}, 3$

The value of the determinant $\left|\begin{array}{ccc}{\sin }^2\left(36°\right)& {\cos }^2\left(36°\right)& \cot \left(135°\right)\\ {\sin }^2\left(53°\right)& \cot \left(135°\right)& {\cos }^2\left(53°\right)\\ \cot \left(135°\right)& {\cos }^2\left(25°\right)& {\cos }^2\left(65°\right)\end{array}\right|$ is

• - 2

• - 1

• 0

• 1

If A = $\left[\begin{array}{ccc}\mathrm{x}& 1& - \mathrm{x}\\ 0& 1& - 1\\ \mathrm{x}& 0& 7\end{array}\right]$ and det(A) = $\left|\begin{array}{ccc}3& 0& 1\\ 2& - 1& 0\\ 0& 6& 7\end{array}\right|$, then value of x is

• - 3

• 3

• 2

• - 8

The coefficient of x² in the expansion of the determinnat $\left|\begin{array}{ccc}{\mathrm{x}}^2& {\mathrm{x}}^3 + 1& {\mathrm{x}}^5 + 2\\ {\mathrm{x}}^3 + 3& {\mathrm{x}}^2 + \mathrm{x}& {\mathrm{x}}^3 + {\mathrm{x}}^4\\ \mathrm{x} + 4& {\mathrm{x}}^3 + {\mathrm{x}}^4& 2^3\end{array}\right|$ is

• - 10

• - 8

• - 2

• - 6

Let A = $\left[\begin{array}{cc}1& \frac{- 1 - \mathrm{i}\sqrt{3}}{2}\\ \frac{- 1 + \mathrm{i}\sqrt{3}}{2}& 1\end{array}\right]$. Then A¹⁰⁰ is equal to

• 2¹⁰⁰A

• 2⁹⁹A

• 2⁹⁸A

• A

If f(x) = $\left|\begin{array}{ccc}\mathrm{x}& {\mathrm{x}}^2& {\mathrm{x}}^3\\ 1& 2\mathrm{x}& 3{\mathrm{x}}^2\\ 0& 2& 6\mathrm{x}\end{array}\right|$, then f'(x) is equal to

• x³ + 6x²

• 6x³

• 3x

• 6x²

If A = $\left|\begin{array}{ccc}4& \mathrm{k}& \mathrm{k}\\ 0& \mathrm{k}& \mathrm{k}\\ 0& 0& \mathrm{k}\end{array}\right|$ and det(A) = 256, then $\left|\mathrm{k}\right|$ equals

• 4

• 5

• 6

• 8

If $\mathrm{A} = \left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right)$, then Aⁿ + nI is equal to

• I

• nA

• I + nA

• I - nA

If A = $\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$, then A²⁰¹⁷ is equal to

• 2²⁰¹⁵A

• 2²⁰¹⁶A

• 2²⁰¹⁴A

• 2²⁰¹⁷A