(高等教育出版)线性代数课后习题1-4作业答案.docVIP

第一章 行列式 1( 利用对角线法则计算下列三阶行列式( (1)( 解 (2(((4)(3(0(((1)(((1)(1(1(8 (0(1(3(2(((1)(8(1(((4)(((1) ((24(8(16(4((4( (3)( 解 (bc2(ca2(ab2(ac2(ba2(cb2 ((a(b)(b(c)(c(a)( 4( 计算下列各行列式( (1)( 解 ( (2)( 解 ( (3)( 解 ( (4)( 解 (abcd(ab(cd(ad(1( 6. 证明: (1)((a(b)3; 证明 ((a(b)3 ( (2); 证明 ( 8. 计算下列各行列式(Dk为k阶行列式)( (1), 其中对角线上元素都是a( 未写出的元素都是0( 解 (按第n行展开) (an(an(2(an(2(a2(1)( (2); 解 将第一行乘((1)分别加到其余各行( 得 ( 再将各列都加到第一列上( 得 ([x((n(1)a](x(a)n 第二章　矩阵及其运算 1. 计算下列乘积( (5)( 解 ((a11x1(a12x2(a13x3 a12x1(a22x2(a23x3 a13x1(a23x2(a33x3) ( 2. 设( ( 求3AB(2A及ATB( 解 ( ( 3. 已知两个线性变换 ( ( 求从z1( z2( z3到x1( x2( x3的线性变换( 解 由已知 ( 所以有( 4. 设( ( 问( (1)AB(BA吗? 解 AB(BA( 因为( ( 所以AB(BA( (3)(A(B)(A(B)(A2(B2吗? 解 (A(B)(A(B)(A2(B2( 因为( ( ( 而 ( 故(A(B)(A(B)(A2(B2( 5. 举反列说明下列命题是错误的( (1)若A2(0( 则A(0( 解 取( 则A2(0( 但A(0( (2)若A2(A( 则A(0或A(E( 解 取( 则A2(A( 但A(0且A(E( (3)若AX(AY( 且A(0( 则X(Y ( 解 取 ( ( ( 则AX(AY( 且A(0( 但X(Y ( 7. 设( 求Ak ( 解 首先观察 ( ( ( ( ( ( ( ( ( (( ( 用数学归纳法证明( 当k(2时( 显然成立( 假设k时成立，则k(1时， ( 由数学归纳法原理知( ( 8. 设A( B为n阶矩阵，且A为对称矩阵，证明BTAB也是对称矩阵( 证明 因为AT(A( 所以 (BTAB)T(BT(BTA)T(BTATB(BTAB( 从而BTAB是对称矩阵( 11( 求下列矩阵的逆矩阵( (1)( 解 ( |A|(1( 故A(1存在( 因为 ( 故 ( (3)( 解 ( |A|(2(0( 故A(1存在( 因为 ( 所以 ( (4)(a1a2( ( (an (0) ( 解 ( 由对角矩阵的性质知 ( 12. 利用逆矩阵解下列线性方程组( (1)( 解 方程组可表示为 ( 故 ( 从而有 ( 19.设P(1AP((( 其中( ( 求A11( 解 由P(1AP((( 得A(P(P(1( 所以A11( A=P(11P(1. |P|(3( ( (