1The absolute value of \( -7 \) is written \( |-7| \). What is it? Are a determinant and an absolute value the same thing?
2If \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), which of these is the value of \( |A| \): \( bc - ad \), \( ad - bc \), \( ac - bd \) or \( bd - ac \)?
3If the determinant of the order one matrix \( A = [-5] \) is written \( |A| = |-5| \), which of these is its value: \( 5 \), \( \frac{1}{5} \), \( -5 \) or \( 5^{-1} \)?
4If \( A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \), which of these is the value of \( |A| \): \( -12 \), \( -5 \), \( 5 \) or \( 10 \)?
5If \( \begin{vmatrix} 2 & x \\ 1 & x \end{vmatrix} = 0 \), which of these is the value of x: 2, 1, \( -2 \) or 0?
6Find the value of the determinant \( |2| \).
7Find the value of the determinant \( |-7| \).
8Find the value of \( \begin{vmatrix} 2 & 3 \\ 1 & 4 \end{vmatrix} \).
9Find the value of \( \begin{vmatrix} \sin A & -\cos A \\ \cos A & \sin A \end{vmatrix} \).
10Find the value of \( \begin{vmatrix} x + y & x - y \\ x - y & x + y \end{vmatrix} \).
11Find the value of \( \begin{vmatrix} x + y & x - y \\ x + y & x - y \end{vmatrix} \).
12Find the value of \( \begin{vmatrix} 3 & -4 \\ -3 & 5 \end{vmatrix} \).
13Find the value of \( \begin{vmatrix} x + y & x - y \\ x^2 + xy + y^2 & x^2 - xy + y^2 \end{vmatrix} \).
14Find the determinant of the matrix \( [10] \).
15Find the determinant of the matrix \( [-8] \).
16Find the determinant of the matrix \( \begin{bmatrix} x & y \\ y & x \end{bmatrix} \).
17Find the determinant of the matrix \( \begin{bmatrix} 1 & -3 \\ 2 & 4 \end{bmatrix} \).
18Find the determinant of the matrix \( \begin{bmatrix} 10 & 5 \\ 4 & 2 \end{bmatrix} \).
19Find the determinant of the matrix \( \begin{bmatrix} -7 & 11 \\ 2 & -4 \end{bmatrix} \).
20Find the value of x if \( \begin{vmatrix} 3 & x \\ 2 & 8 \end{vmatrix} = 20 \).
21Find the value of x if \( \begin{vmatrix} 5 & -2 \\ 2x & -3 \end{vmatrix} = 1 \).
22Find the value of x if \( \begin{vmatrix} 3x & 2 \\ -4 & 3 \end{vmatrix} = 8 \).
23Find the value of x if \( \begin{vmatrix} x & -2x \\ 3x & 4x \end{vmatrix} = 100 \).
24If \( A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & -5 \\ 6 & 2 \end{bmatrix} \), find the determinant of \( A - 3B \).
25With the same A and B, find the determinant of \( 2A + B \).
26With the same A and B, find the determinant of \( 3A - 2B \).
27With the same A and B, find the determinant of \( A + 2B \).
28Prove that \( \begin{vmatrix} x^2 & y^2 \\ y & x \end{vmatrix} = (x - y)(x^2 + xy + y^2) \).
29Prove that \( \begin{vmatrix} x + y & y + z \\ z - y & x - y \end{vmatrix} = x^2 - z^2 \).
30Prove that \( \begin{vmatrix} a - 1 & a^2 - a + 1 \\ a + 1 & a^2 + a + 1 \end{vmatrix} = -2 \).
31Prove that \( \begin{vmatrix} 1 + x & 1 \\ 1 & 1 + y \end{vmatrix} = xy\left(1 + \frac{1}{x} + \frac{1}{y}\right) \).
32If \( M = \begin{bmatrix} 5 & 2 \\ 4 & 3 \end{bmatrix} \) and \( N = \begin{bmatrix} 1 & -3 \\ 6 & 2 \end{bmatrix} \), verify that \( |MN| = |N||M| \).
33Why does the matrix \( \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \) have no determinant?
34Show that \( \begin{bmatrix} 6 & 4 \\ 9 & 6 \end{bmatrix} \) is a singular matrix.
35Is \( \begin{bmatrix} -2 & 3 \\ 1 & 4 \end{bmatrix} \) singular or non-singular? Give a reason.