【课后习题】线性代数第六版第二章矩阵及其运算习题二

习题二

1. 计算下列乘积:

(1) ( 4 3 1 1 − 2 3 5 7 0 ) ( 7 2 1 ) \left(\begin{array}{rrr}4 & 3 & 1 \\ 1 & -2 & 3 \\ 5 & 7 & 0\end{array}\right)\left(\begin{array}{l}7 \\ 2 \\ 1\end{array}\right) ⎝⎛4153−27130⎠⎞⎝⎛721⎠⎞;

(2) ( 1 , 2 , 3 ) ( 3 2 1 ) (1,2,3)\left(\begin{array}{l}3 \\ 2 \\ 1\end{array}\right) (1,2,3)⎝⎛321⎠⎞;

(3) ( 2 1 3 ) ( − 1 , 2 ) \left(\begin{array}{l}2 \\ 1 \\ 3\end{array}\right)(-1,2) ⎝⎛213⎠⎞(−1,2);

(4) ( 2 1 4 0 1 − 1 3 4 ) ( 1 3 1 0 − 1 2 1 − 3 1 4 0 − 2 ) \left(\begin{array}{rrrr}2 & 1 & 4 & 0 \\ 1 & -1 & 3 & 4\end{array}\right)\left(\begin{array}{rrr}1 & 3 & 1 \\ 0 & -1 & 2 \\ 1 & -3 & 1 \\ 4 & 0 & -2\end{array}\right) (211−14304)⎝⎜⎜⎛10143−1−30121−2⎠⎟⎟⎞;

(5) ( x 1 , x 2 , x 3 ) ( a 11 a 12 a 13 a 12 a 22 a 23 a 13 a 23 a 33 ) ( x 1 x 2 x 3 ) \left(x_1, x_2, x_3\right)\left(\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33}\end{array}\right)\left(\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right) (x1,x2,x3)⎝⎛a11a12a13a12a22a23a13a23a33⎠⎞⎝⎛x1x2x3⎠⎞.

2. 设 A = ( 1 1 1 1 1 − 1 1 − 1 1 ) , B = ( 1 2 3 − 1 − 2 4 0 5 1 ) \boldsymbol{A}=\left(\begin{array}{rrr}1 & 1 & 1 \\ 1 & 1 & -1 \\ 1 & -1 & 1\end{array}\right), \boldsymbol{B}=\left(\begin{array}{rrr}1 & 2 & 3 \\ -1 & -2 & 4 \\ 0 & 5 & 1\end{array}\right) A=⎝⎛11111−11−11⎠⎞,B=⎝⎛1−102−25341⎠⎞, 求 3 A B − 2 A 3 \boldsymbol{A B}-2 \boldsymbol{A} 3AB−2A 及 A T B \boldsymbol{A}^{\mathrm{T}} \boldsymbol{B} ATB.

3. 已知两个线性变换

{ x 1 = 2 y 1 + y 3 , x 2 = − 2 y 1 + 3 y 2 + 2 y 3 , x 3 = 4 y 1 + y 2 + 5 y 3 , { y 1 = − 3 z 1 + z 2 , y 2 = 2 z 1 + z 3 , y 3 = − z 2 + 3 z 3 , \left\{\begin{array} { l } { x _ { 1 } = 2 y _ { 1 } + y _ { 3 } , } \\ { x _ { 2 } = - 2 y _ { 1 } + 3 y _ { 2 } + 2 y _ { 3 } , } \\ { x _ { 3 } = 4 y _ { 1 } + y _ { 2 } + 5 y _ { 3 } , } \end{array} \left\{\begin{array}{l} y_1=-3 z_1+z_2, \\ y_2=2 z_1+z_3, \\ y_3=-z_2+3 z_3, \end{array}\right.\right. ⎩⎨⎧x1=2y1+y3,x2=−2y1+3y2+2y3,x3=4y1+y2+5y3,⎩⎨⎧y1=−3z1+z2,y2=2z1+z3,y3=−z2+3z3,
求从 z 1 , z 2 , z 3 z_1, z_2, z_3 z1,z2,z3 到 x 1 , x 2 , x 3 x_1, x_2, x_3 x1,x2,x3 的线性变换.

4. 设 A = ( 1 2 1 3 ) , B = ( 1 0 1 2 ) \boldsymbol{A}=\left(\begin{array}{ll}1 & 2 \\ 1 & 3\end{array}\right), \boldsymbol{B}=\left(\begin{array}{ll}1 & 0 \\ 1 & 2\end{array}\right) A=(1123),B=(1102), 问 :

(1) A B = B A \boldsymbol{A} \boldsymbol{B}=\boldsymbol{B} \boldsymbol{A} AB=BA 吗?

(2) ( A + B ) 2 = A 2 + 2 A B + B 2 (A+B)^2=A^2+2 A B+B^2 (A+B)2=A2+2AB+B2 吗?

(3) ( A + B ) ( A − B ) = A 2 − B 2 (\boldsymbol{A}+\boldsymbol{B})(\boldsymbol{A}-\boldsymbol{B})=\boldsymbol{A}^2-\boldsymbol{B}^2 (A+B)(A−B)=A2−B2 吗?

5. 举反例说明下列命题是错误的:

(1) 若 A 2 = O \boldsymbol{A}^2=\boldsymbol{O} A2=O, 则 A = O \boldsymbol{A}=\boldsymbol{O} A=O;

(2) 若 A 2 = A \boldsymbol{A}^2=\boldsymbol{A} A2=A, 则 A = O \boldsymbol{A}=\boldsymbol{O} A=O 或 A = E \boldsymbol{A}=\boldsymbol{E} A=E;

(3) 若 A X = A Y \boldsymbol{A} \boldsymbol{X}=\boldsymbol{A} \boldsymbol{Y} AX=AY, 且 A ≠ O \boldsymbol{A} \neq \boldsymbol{O} A=O, 则 X = Y \boldsymbol{X}=\boldsymbol{Y} X=Y.

6. （1）设 A = ( 1 0 λ 1 ) \boldsymbol{A}=\left(\begin{array}{ll}1 & 0 \\ \lambda & 1\end{array}\right) A=(1λ01), 求 A 2 , A 3 , ⋯ , A k ; \boldsymbol{A}^2, \boldsymbol{A}^3, \cdots, \boldsymbol{A}^k ; \quad A2,A3,⋯,Ak; （2）设 A = ( λ 1 0 0 λ 1 0 0 λ ) \boldsymbol{A}=\left(\begin{array}{ccc}\lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda\end{array}\right) A=⎝⎛λ001λ001λ⎠⎞, 求 A 4 \boldsymbol{A}^4 A4.

7. (1) 设 A = ( 3 1 1 − 3 ) \boldsymbol{A}=\left(\begin{array}{rr}3 & 1 \\ 1 & -3\end{array}\right) A=(311−3), 求 A 50 \boldsymbol{A}^{50} A50 和 A 51 \boldsymbol{A}^{51} A51;

（2）设 a = ( 2 1 − 3 ) , b = ( 1 2 4 ) , A = a b T \boldsymbol{a}=\left(\begin{array}{r}2 \\ 1 \\ -3\end{array}\right), \boldsymbol{b}=\left(\begin{array}{l}1 \\ 2 \\ 4\end{array}\right), \boldsymbol{A}=\boldsymbol{a} \boldsymbol{b}^{\mathrm{T}} a=⎝⎛21−3⎠⎞,b=⎝⎛124⎠⎞,A=abT, 求 A 100 \boldsymbol{A}^{100} A100.

8. （1）设 A , B \boldsymbol{A}, \boldsymbol{B} A,B 为 n n n 阶矩阵, 且 A \boldsymbol{A} A 为对称矩阵,证明 B T A B \boldsymbol{B}^{\mathrm{T}} \boldsymbol{A B} BTAB 也是对称矩阵;

(2) 设 A , B \boldsymbol{A}, \boldsymbol{B} A,B 都是 n n n 阶对称矩阵,证明 A B \boldsymbol{A B} AB 是对称矩阵的充分必要条件是 A B = B A \boldsymbol{A B}=\boldsymbol{B A} AB=BA.

9. 求下列矩阵的逆矩阵:

(1) ( 1 2 2 5 ) \left(\begin{array}{ll}1 & 2 \\ 2 & 5\end{array}\right) (1225);

(2) ( cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ ) \left(\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right) (cosθsinθ−sinθcosθ)

(3) ( 1 2 − 1 3 4 − 2 5 − 4 1 ) \left(\begin{array}{rrr}1 & 2 & -1 \\ 3 & 4 & -2 \\ 5 & -4 & 1\end{array}\right) ⎝⎛13524−4−1−21⎠⎞;

(4) ( a 1 0 a 2 ⋱ 0 a n ) ( a 1 a 2 ⋯ a n ≠ 0 ) \left(\begin{array}{cccc}a_1 & & & 0 \\ & a_2 & & \\ & & \ddots & \\ 0 & & & a_n\end{array}\right)\left(a_1 a_2 \cdots a_n \neq 0\right) ⎝⎜⎜⎛a10a2⋱0an⎠⎟⎟⎞(a1a2⋯an=0).

10. 已知线性变换

{ x 1 = 2 y 1 + 2 y 2 + y 3 , x 2 = 3 y 1 + y 2 + 5 y 3 , x 3 = 3 y 1 + 2 y 2 + 3 y 3 , \left\{\begin{array}{l} x_1=2 y_1+2 y_2+y_3, \\ x_2=3 y_1+y_2+5 y_3, \\ x_3=3 y_1+2 y_2+3 y_3, \end{array}\right. ⎩⎨⎧x1=2y1+2y2+y3,x2=3y1+y2+5y3,x3=3y1+2y2+3y3,
求从变量 x 1 , x 2 , x 3 x_1, x_2, x_3 x1,x2,x3 到变量 y 1 , y 2 , y 3 y_1, y_2, y_3 y1,y2,y3 的线性变换.

11. 设 J \boldsymbol{J} J 是元素全为 1 的 n ( ⩾ 2 ) n(\geqslant 2) n(⩾2) 阶方阵. 证明 E − J \boldsymbol{E}-\boldsymbol{J} E−J 是可逆方阵, 且 ( E − J ) − 1 = E − 1 n − 1 J (\boldsymbol{E}-\boldsymbol{J})^{-1}=\boldsymbol{E}-\frac{1}{n-1} \boldsymbol{J} (E−J)−1=E−n−11J, 这里 E \boldsymbol{E} E 是与 J \boldsymbol{J} J 同阶的单位矩阵.

12. 设 A k = O \boldsymbol{A}^k=\boldsymbol{O} Ak=O ( k k k 为正整数 ), 证明

( E − A ) − 1 = E + A + A 2 + ⋯ + A h − 1 . (\boldsymbol{E}-\boldsymbol{A})^{-1}=\boldsymbol{E}+\boldsymbol{A}+\boldsymbol{A}^2+\cdots+\boldsymbol{A}^{h-1} . (E−A)−1=E+A+A2+⋯+Ah−1.

13. 设方阵 A \boldsymbol{A} A 满足 A 2 − A − 2 E = O \boldsymbol{A}^2-\boldsymbol{A}-2 \boldsymbol{E}=\boldsymbol{O} A2−A−2E=O, 证明 A \boldsymbol{A} A 及 A + 2 E \boldsymbol{A}+2 \boldsymbol{E} A+2E 都可逆, 并求 A − 1 \boldsymbol{A}^{-1} A−1 及 ( A + 2 E ) − 1 (\boldsymbol{A}+2 \boldsymbol{E})^{-1} (A+2E)−1.

14. 解下列矩阵方程 :

(1) ( 2 5 1 3 ) X = ( 4 − 6 2 1 ) \left(\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right) \boldsymbol{X}=\left(\begin{array}{rr}4 & -6 \\ 2 & 1\end{array}\right) (2153)X=(42−61)

(2) X ( 2 1 − 1 2 1 0 1 − 1 1 ) = ( 1 − 1 3 4 3 2 ) X\left(\begin{array}{rrr}2 & 1 & -1 \\ 2 & 1 & 0 \\ 1 & -1 & 1\end{array}\right)=\left(\begin{array}{rrr}1 & -1 & 3 \\ 4 & 3 & 2\end{array}\right) X⎝⎛22111−1−101⎠⎞=(14−1332);

(3) ( 1 4 − 1 2 ) X ( 2 0 − 1 1 ) = ( 3 1 0 − 1 ) \left(\begin{array}{rr}1 & 4 \\ -1 & 2\end{array}\right) \boldsymbol{X}\left(\begin{array}{rr}2 & 0 \\ -1 & 1\end{array}\right)=\left(\begin{array}{rr}3 & 1 \\ 0 & -1\end{array}\right) (1−142)X(2−101)=(301−1);

(4) A X B = C \boldsymbol{A} \boldsymbol{X} \boldsymbol{B}=\boldsymbol{C} AXB=C, 其中 A = ( 2 1 5 4 ) , B = ( 1 3 3 1 4 3 1 3 4 ) , C = ( 1 0 − 1 1 − 2 0 ) \boldsymbol{A}=\left(\begin{array}{ll}2 & 1 \\ 5 & 4\end{array}\right), \boldsymbol{B}=\left(\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right), \boldsymbol{C}=\left(\begin{array}{rrr}1 & 0 & -1 \\ 1 & -2 & 0\end{array}\right) A=(2514),B=⎝⎛111343334⎠⎞,C=(110−2−10).

15. 分别应用克拉默法则和逆矩阵解下列线性方程组:

(1) { x 1 + 2 x 2 + 3 x 3 = 1 , 2 x 1 + 2 x 2 + 5 x 3 = 2 3 x 1 + 5 x 2 + x 3 = 3 ; \left\{\begin{array}{l}x_1+2 x_2+3 x_3=1, \\ 2 x_1+2 x_2+5 x_3=2 \\ 3 x_1+5 x_2+x_3=3 ;\end{array}\right. ⎩⎨⎧x1+2x2+3x3=1,2x1+2x2+5x3=23x1+5x2+x3=3;

(2) { x 1 + x 2 + x 3 = 2 , x 1 + 2 x 2 + 4 x 3 = 3 , x 1 + 3 x 2 + 9 x 3 = 5. \left\{\begin{array}{l}x_1+x_2+x_3=2, \\ x_1+2 x_2+4 x_3=3, \\ x_1+3 x_2+9 x_3=5 .\end{array}\right. ⎩⎨⎧x1+x2+x3=2,x1+2x2+4x3=3,x1+3x2+9x3=5.

16. 设 A \boldsymbol{A} A 为 3 阶矩阵, ∣ A ∣ = 1 2 |\boldsymbol{A}|=\frac{1}{2} ∣A∣=21, 求 ∣ ( 2 A ) − 1 − 5 A ∗ ∣ \left|(2 \boldsymbol{A})^{-1}-5 \boldsymbol{A}^*\right| ∣∣(2A)−1−5A∗∣∣.

17. 设 A = ( 0 3 3 1 1 0 − 1 2 3 ) , A B = A + 2 B \boldsymbol{A}=\left(\begin{array}{rrr}0 & 3 & 3 \\ 1 & 1 & 0 \\ -1 & 2 & 3\end{array}\right), \boldsymbol{A B}=\boldsymbol{A}+2 \boldsymbol{B} A=⎝⎛01−1312303⎠⎞,AB=A+2B, 求 B \boldsymbol{B} B.

18. 设 A = ( 1 0 1 0 2 0 1 0 1 ) \boldsymbol{A}=\left(\begin{array}{lll}1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 1\end{array}\right) A=⎝⎛101020101⎠⎞, 且 A B + E = A 2 + B \boldsymbol{A} \boldsymbol{B}+\boldsymbol{E}=\boldsymbol{A}^2+\boldsymbol{B} AB+E=A2+B, 求 B \boldsymbol{B} B.

19. 设 A = diag ⁡ ( 1 , − 2 , 1 ) , A ∗ B A = 2 B A − 8 E \boldsymbol{A}=\operatorname{diag}(1,-2,1), \boldsymbol{A}{ }^* \boldsymbol{B} \boldsymbol{A}=2 \boldsymbol{B} \boldsymbol{A}-8 \boldsymbol{E} A=diag(1,−2,1),A∗BA=2BA−8E, 求 B \boldsymbol{B} B.

20. 已知矩阵 A \boldsymbol{A} A 的伴随矩阵 A ∗ = diag ⁡ ( 1 , 1 , 1 , 8 ) \boldsymbol{A}^*=\operatorname{diag}(1,1,1,8) A∗=diag(1,1,1,8), 且 A B A − 1 = B A − 1 + 3 E \boldsymbol{A} \boldsymbol{B} \boldsymbol{A}^{-1}=\boldsymbol{B} \boldsymbol{A}^{-1}+3 \boldsymbol{E} ABA−1=BA−1+3E, 求 B \boldsymbol{B} B.

21. 设 P − 1 A P = Λ \boldsymbol{P}^{-1} \boldsymbol{A P}=\boldsymbol{\Lambda} P−1AP=Λ, 其中 P = ( − 1 − 4 1 1 ) , Λ = ( − 1 0 0 2 ) \boldsymbol{P}=\left(\begin{array}{rr}-1 & -4 \\ 1 & 1\end{array}\right), \boldsymbol{\Lambda}=\left(\begin{array}{rr}-1 & 0 \\ 0 & 2\end{array}\right) P=(−11−41),Λ=(−1002), 求 A 11 \boldsymbol{A}^{11} A11.

22. 设 A P = P Λ \boldsymbol{A P}=\boldsymbol{P} \boldsymbol{\Lambda} AP=PΛ, 其中

P = ( 1 1 1 1 0 − 2 1 − 1 1 ) , Λ = ( − 1 1 5 ) , 求 φ ( A ) = A 8 ( 5 E − 6 A + A 2 ) . \boldsymbol{P}=\left(\begin{array}{rrr} 1 & 1 & 1 \\ 1 & 0 & -2 \\ 1 & -1 & 1 \end{array}\right), \boldsymbol{\Lambda}=\left(\begin{array}{lll} -1 & & \\ & 1 & \\ & & 5 \end{array}\right) \text {, 求 } \varphi(\boldsymbol{A})=\boldsymbol{A}^8\left(5 \boldsymbol{E}-6 \boldsymbol{A}+\boldsymbol{A}^2\right) \text {. } P=⎝⎛11110−11−21⎠⎞,Λ=⎝⎛−115⎠⎞, 求 φ(A)=A8(5E−6A+A2).

23. 设矩阵 A \boldsymbol{A} A 可逆, 证明其伴随矩阵 A ∗ A^* A∗ 也可逆, 且 ( A ∗ ) − 1 = ( A − 1 ) ∗ \left(A^\right)^{-1}=\left(A^{-1}\right)^ (A∗)−1=(A−1)∗.

24. 设 n n n 阶矩阵 A \boldsymbol{A} A 的伴随矩阵为 A ∗ \boldsymbol{A}^* A∗, 证明:

(1) 若 ∣ A ∣ = 0 |\boldsymbol{A}|=0 ∣A∣=0,则 ∣ A ∗ ∣ = 0 \left|\boldsymbol{A}^*\right|=0 ∣A∗∣=0;

(2) ∣ A ∗ ∣ = ∣ A ∣ n − 1 \left|\boldsymbol{A}^*\right|=|\boldsymbol{A}|^{n-1} ∣A∗∣=∣A∣n−1.

25. 计算 ( 1 2 1 0 0 1 0 1 0 0 2 1 0 0 0 3 ) ( 1 0 3 1 0 1 2 − 1 0 0 − 2 3 0 0 0 − 3 ) \left(\begin{array}{llll}1 & 2 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 3\end{array}\right)\left(\begin{array}{rrrr}1 & 0 & 3 & 1 \\ 0 & 1 & 2 & -1 \\ 0 & 0 & -2 & 3 \\ 0 & 0 & 0 & -3\end{array}\right) ⎝⎜⎜⎛1000210010200113⎠⎟⎟⎞⎝⎜⎜⎛1000010032−201−13−3⎠⎟⎟⎞.

26. 设 A = ( 3 4 0 0 4 − 3 0 0 0 0 2 0 0 0 2 2 ) \boldsymbol{A}=\left(\begin{array}{rrrr}3 & 4 & 0 & 0 \\ 4 & -3 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 2 & 2\end{array}\right) A=⎝⎜⎜⎛34004−30000220002⎠⎟⎟⎞, 求 ∣ A 8 ∣ \left|\boldsymbol{A}^8\right| ∣∣A8∣∣ 及 A 4 \boldsymbol{A}^4 A4.

27. 设 n n n 阶矩阵 A \boldsymbol{A} A 及 s s s 阶矩阵 B \boldsymbol{B} B 都可逆,求 ( O A B O ) − 1 \left(\begin{array}{ll}\boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O}\end{array}\right)^{-1} (OBAO)−1.

28. 求下列矩阵的逆矩阵:

(1) ( 5 2 0 0 2 1 0 0 0 0 8 3 0 0 5 2 ) \left(\begin{array}{llll}5 & 2 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 0 & 0 & 8 & 3 \\ 0 & 0 & 5 & 2\end{array}\right) ⎝⎜⎜⎛5200210000850032⎠⎟⎟⎞;

( 2 ) ( 0 0 1 5 2 1 0 4 3 0 ) (2)\left(\begin{array}{lll}0 & 0 & \frac{1}{5} \\ 2 & 1 & 0 \\ 4 & 3 & 0\end{array}\right) (2)⎝⎛0240135100⎠⎞.