5.10 复习题 $$1 \text{ (a)} \quad \begin{bmatrix} 12 & 18 & -40 \ 0 & 0 & 8 \ 0 & 0 & 4 \end{bmatrix} \begin{bmatrix} 12 & 0 & 0 \ 18 & 0 & 0 \ -40 & 8 & 4 \end{bmatrix}$$

$$\text{(b)} \quad \begin{bmatrix} 8 & 9 & -7 \ 0 & 2 & 0 \ 0 & 0 & 5 \end{bmatrix} \begin{bmatrix} 12 & 12 & -6 \ 0 & 4 & 0 \ 0 & 0 & 16 \end{bmatrix}$$

$$\begin{bmatrix} 4 & 14 & -3 \ 0 & 0 & 5 \ 0 & 0 & 4 \end{bmatrix}$$

$$2 \quad \lambda = -1, \mu = 2 \ \lambda = 2, \mu = -1$$

$$3 \text{ Normal strain} = \begin{bmatrix} 3 \ -3 \ 3\sqrt{2} \end{bmatrix}$$

$$\text{Shear strain} = 0$$

$$4 \quad (\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)(\alpha + \beta + \gamma)$$

$$5 \quad \theta = 1: (1 + 2\alpha, -3\alpha, \alpha) \ \theta = 2: (2\alpha, 1 - 3\alpha, \alpha)$$

$$6 \quad \mathbf{A}^2 = \begin{bmatrix} 1 & 3 & 2 \ 0 & 5 & 2 \ 2 & -2 & 1 \end{bmatrix} \quad \mathbf{A}^3 = \begin{bmatrix} 3 & 6 & 5 \ 6 & 7 & 8 \ -4 & 1 & -3 \end{bmatrix}$$

$$\mathbf{A}^{-1} = \begin{bmatrix} 3 & -2 & -1 \ 2 & -1 & 0 \ -4 & 3 & 1 \end{bmatrix} \quad \mathbf{X} = \begin{bmatrix} -11 \ -1 \ 15 \end{bmatrix}$$

$$7 \text{ (a)} \quad \mathbf{P}^T = \frac{1}{3} \begin{bmatrix} 2 & -2 & -1 \ 1 & 2 & -2 \ 2 & 1 & 2 \end{bmatrix}, \text{ the solution } x = \mathbf{P}^{-1}\mathbf{b}$$

exists

$$\text{(b)} \quad \mathbf{E} = \begin{bmatrix} I_x - \frac{Q_x^2}{A} & I_{xy} - \frac{Q_x Q_y}{A} & 0 \ I_{xy} - \frac{Q_x Q_y}{A} & I_y - \frac{Q_y^2}{A} & 0 \ 0 & 0 & A \end{bmatrix}$$

$$8 \text{ (a)} \quad \mathbf{B} = \begin{bmatrix} 8 & -10 & 3 \ 3 & -4 & 1 \ -1 & 0 & 0 \end{bmatrix}$$

$$\text{(b)} \quad k = 8.2316, k = -1.9316$$

$$9 \text{ (a)} \quad \begin{bmatrix} -2 & -3 & -1 \ -8 & -1 & 29 \ -6 & 2 & 8 \end{bmatrix}, -22, \frac{1}{22} \begin{bmatrix} 2 & 3 & 1 \ 8 & 1 & -29 \ 6 & -2 & -8 \end{bmatrix},$$

$$z = 2, y = 1, x = 2$$

$$\text{(b)} \quad \mathbf{Y} = \begin{bmatrix} 8 & 1 & 9 \ 18 & 3 & -35 \ 8 & -4 & -6 \end{bmatrix}$$

$$\text{(c)} \quad \mathbf{Z} = \begin{bmatrix} 6 & 11 & 3 \ 15 & 2 & -59 \ 16 & -7 & -16 \end{bmatrix}$$

$$10 \text{ (a)} \quad 3, 0, 2, 1; \det = 12$$

$$\text{(b)} \quad 1, 2, 3, 4$$

$$11 \quad 1, 2, 3$$

$$12 \text{ If } c \neq 0 \text{ then rank} = 2 \ \text{if } c = 0 \text{ then rank} = 1$$

$$13 \quad \mathbf{a} = \begin{bmatrix} 0.0051 \ 0.9712 \ -0.3931 \ -0.0760 \ 0.0283 \end{bmatrix} \quad f(2) = 0.2200 \quad f(3.5) = -0.4228$$

$$14 \quad a = 0.4424, b = -1.5037, c = 1.5023, d = -0.0611 \ \text{max at } x = 0.74, f = 0.4065$$

$$15 \quad \text{rank } \mathbf{B} = 2, \mathbf{AA}^T = \mathbf{I}, \mathbf{A}^{-1} = \mathbf{A}^T \ x_1 = 2.444, x_2 = -2.556, x_3 = -1.222$$

$$16 \text{ (b)} \quad x_1 = 44, x_2 = -48, x_3 = -39, x_4 = 33$$

$$17 \text{ (a)} \quad \begin{bmatrix} 0 \ -1 \ 2 \end{bmatrix} \quad \text{(b)} \quad \begin{bmatrix} -8 \ -5 \ \frac{1}{2} \end{bmatrix}$$

$$18 \text{ (a)} \quad 4, 3, 2; \begin{bmatrix} 0.5774 \ 0.5774 \ -0.5774 \end{bmatrix}, \begin{bmatrix} 0.1961 \ 0.5883 \ -0.7845 \end{bmatrix}, \begin{bmatrix} 0 \ 0.7071 \ -0.7071 \end{bmatrix}$$

$$\text{(b)} \quad 5, 3, -1; \begin{bmatrix} 0.2033 \ 0.6505 \ 0.7318 \end{bmatrix}, \begin{bmatrix} 0.1374 \ 0.8242 \ 0.5494 \end{bmatrix}, \begin{bmatrix} -0.4472 \ -0.8944 \ 0 \end{bmatrix}$$

$$\text{(c)} \quad 9, 6, 3; \quad \frac{1}{3} \begin{bmatrix} -1 \ 2 \ 2 \end{bmatrix}, \frac{1}{3} \begin{bmatrix} 2 \ -1 \ 2 \end{bmatrix}, \frac{1}{3} \begin{bmatrix} 2 \ 2 \ -1 \end{bmatrix}$$

$$19 \quad \lambda = 9 \quad \alpha = 1 \quad \beta = 6$$