$$25 \mathbf{A} = \begin{bmatrix} 1 & \frac{5}{2} & \frac{3}{2} \ \frac{5}{2} & -1 & 2 \ \frac{3}{2} & 2 & 1 \end{bmatrix} + \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} \ -\frac{1}{2} & 0 & -2 \ -\frac{1}{2} & 2 & 0 \end{bmatrix}$$

$$26 \begin{bmatrix} 41 & 15 & 7 \ -9 & 63 & -40 \ -13 & 38 & 41 \end{bmatrix}$$

$$27 £2273.88$$

$$28 \begin{bmatrix} 1 & 3 & 2 \ 0 & 5 & 2 \ 2 & -2 & 1 \end{bmatrix} \begin{bmatrix} 3 & 6 & 5 \ 6 & 7 & 8 \ -4 & 1 & -3 \end{bmatrix}$$

$$30 \begin{bmatrix} 799.8 \ 800 \ 800.2 \end{bmatrix} \begin{bmatrix} 800 \ 800 \ 800 \end{bmatrix}$$

$$31 h = \frac{1}{3}, k = \frac{2}{3}, l = \frac{1}{3}, m = \frac{1}{6}$$

$$32 \mathbf{A} = \begin{bmatrix} \sqrt{\frac{1}{2}} & -\sqrt{\frac{1}{2}} \ \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \end{bmatrix}$$

$$\begin{bmatrix} a \ b \end{bmatrix} = \begin{bmatrix} 16 \ -10 \end{bmatrix} + \mathbf{A} \begin{bmatrix} 16 \ -10 \end{bmatrix} \text{ and } \mathbf{B} = \mathbf{A}$$

$$33 n = 3$$

$$34 \text{Minors} = \begin{bmatrix} -1 & 0 & 1 \ -1 & -2 & -1 \ 2 & -2 & -2 \end{bmatrix}$$

$$\text{Cofactors} = \begin{bmatrix} -1 & 0 & 1 \ 1 & -2 & 1 \ 2 & 2 & -2 \end{bmatrix}$$

$$|\mathbf{A}| = 2$$

$$35 \text{ (a) } -19 \quad \text{(b) } 130 \quad \text{(c) } -65$$

$$\text{(d) } 1 \quad \text{(e) } -3$$

$$36 -4, 16, 16, -32$$

$$37 3$$

$$38 \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$$

$$39 \begin{bmatrix} 2 & -1 & 0 \ -4 & 3 & -1 \ 1 & -1 & 1 \end{bmatrix}$$

$$40 2, \begin{bmatrix} 1 & 0 \ -3 & 2 \end{bmatrix}, \frac{1}{2} \begin{bmatrix} 1 & 0 \ -3 & 2 \end{bmatrix}, \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$$

$$44 \text{ (a) } -1.6569, 9.6569$$

$$\text{(b) } 4.6667 \pm j0.62361$$

$$\text{(c) } 2, 3 \pm j$$

$$45 \text{ (a) } -0.1884 \quad \text{(b) } 100$$

$$47 x^2(2x + 1)^2(x - 1)^2$$

$$51 \text{Non-singular, singular, non-singular, singular}$$

$$\begin{bmatrix} -\frac{1}{3} & \frac{2}{3} \ \frac{2}{3} & -\frac{1}{3} \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & -1 \ 0 & 1 & 0 & -1 \ 0 & 0 & 1 & -1 \ 0 & 0 & 0 & 1 \end{bmatrix}$$

$$52 \begin{bmatrix} 1 & -1 & -1 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & \frac{1}{2} & 0 & 0 \ 0 & 0 & \frac{1}{3} & 0 \ 0 & 0 & 0 & \frac{1}{4} \end{bmatrix},$$

$$\begin{bmatrix} 2 & -j \ j & 1 \end{bmatrix}, \frac{1}{3} \begin{bmatrix} 5 & -7 & -1 \ -4 & 5 & 2 \ 2 & -1 & -1 \end{bmatrix}$$

$$53 \frac{1}{169} \begin{bmatrix} -100 & 98 & -92 \ -47 & -79 & -50 \ 6 & -87 & -8 \end{bmatrix}$$

$$55 \begin{bmatrix} 1 & 0 & 0 \ -1 & 1 & 0 \ 2 & -2 & 1 \end{bmatrix}, \frac{1}{4} \begin{bmatrix} 4 & -8 & 6 \ 0 & 2 & -1 \ 0 & 0 & 2 \end{bmatrix}, \frac{1}{4} \begin{bmatrix} 24 & -20 & 3 \ -4 & 4 & -1 \ 4 & -4 & 2 \end{bmatrix}$$

$$56 \begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 2 \ 0 & 1 & 3 \end{bmatrix}, \begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 2 \ 0 & 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix},$$

$$\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & -3 & 2 \ 2 & 3 & -2 \ -1 & -1 & 1 \end{bmatrix}$$

$$57 \frac{1}{5} \begin{bmatrix} -3 & 2 & 2 \ 2 & -3 & 2 \ 2 & 2 & -3 \end{bmatrix}, \begin{bmatrix} 0.68 & -0.32 & -0.32 \ -0.32 & 0.68 & -0.32 \ -0.32 & -0.32 & 0.68 \end{bmatrix}$$

$$58 \frac{1}{68} \begin{bmatrix} -4 & 4 & 8 \ 6 & 11 & -12 \ 36 & -2 & -4 \end{bmatrix}, \frac{1}{49} \begin{bmatrix} -5 & 22 & 8 \ 11 & -19 & 2 \ 13 & -18 & -11 \end{bmatrix}$$