Evaluate the determinant:
\[ \left| \begin{matrix} 2 & 4 \\ -5 & -1 \end{matrix} \right| \]
18
Evaluate the determinants:
(i) \[ \left| \begin{matrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{matrix} \right| \]
(ii) \[ \left| \begin{matrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{matrix} \right| \]
(i) 1
(ii) \(x^3 - x^2 + 2\)
If \(A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}\), then show that \(|2A| = 4|A|\).
If \(A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}\), then show that \(|3A| = 27|A|\).
Evaluate the determinants:
(i) \[ \left| \begin{matrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{matrix} \right| \]
(ii) \[ \left| \begin{matrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{matrix} \right| \]
(iii) \[ \left| \begin{matrix} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{matrix} \right| \]
(iv) \[ \left| \begin{matrix} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{matrix} \right| \]
(i) -12
(ii) 46
(iii) 0
(iv) 5
If \(A = \begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix}\), find \(|A|\).
0
Find values of \(x\), if
(i) \[ \left| \begin{matrix} 2 & 4 \\ 5 & 1 \end{matrix} \right| = \left| \begin{matrix} 2x & 4 \\ 6 & x \end{matrix} \right| \]
(ii) \[ \left| \begin{matrix} 2 & 3 \\ 4 & 5 \end{matrix} \right| = \left| \begin{matrix} x & 3 \\ 2x & 5 \end{matrix} \right| \]
(i) \(x = \pm \sqrt{3}\)
(ii) \(x = 2\)
If
\[ \left| \begin{matrix} x & 2 \\ 18 & x \end{matrix} \right| = \left| \begin{matrix} 6 & 2 \\ 18 & 6 \end{matrix} \right| \]
then \(x\) is equal to
(A) 6
(B) \(\pm 6\)
(C) -6
(D) 0
(B)