线性代数第二章课后答案.docVIP

第二章　矩阵及其运算 1( 已知线性变换( ( 求从变量x1( x2( x3到变量y1( y2( y3的线性变换( 解 由已知( ( 故 ( ( 2( 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( 3( 设( ( 求3AB(2A及ATB( 解 ( ( 4( 计算下列乘积( (1)( 解 ( (2)( 解 ((1(3(2(2(3(1)((10)( (3)( 解 ( (4) ( 解 ( (5)( 解 ((a11x1(a12x2(a13x3 a12x1(a22x2(a23x3 a13x1(a23x2(a33x3) ( 5( 设( ( 问( (1)AB(BA吗? 解 AB(BA( 因为( ( 所以AB(BA( (2)(A(B)2(A2(2AB(B2吗? 解 (A(B)2(A2(2AB(B2( 因为( ( 但 ( 所以(A(B)2(A2(2AB(B2( (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( 6( 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A2( A3( ( ( (( Ak( 解 ( ( ( ( ( ( ( (( ( 8( 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( ( 9( 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( 10( 设A( B都是n阶对称矩阵，证明AB是对称矩阵的充分必要条件是AB(BA( 证明 充分性( 因为AT(A( BT(B( 且AB(BA( 所以 (AB)T((BA)T(ATBT(AB( 即AB是对称矩阵( 必要性( 因为AT(A( BT(B( 且(AB)T(AB( 所以 AB((AB)T(BTAT(BA( 11( 求下列矩阵的逆矩阵( (1)( 解 ( |A|(1( 故A(1存在( 因为 ( 故 ( (2)( 解 ( |A|(1(0( 故A(1存在( 因为 ( 所以 ( (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( 解下列矩阵方程( (1)( 解 ( (2)( 解 ( (3)( 解 ( (4)( 解 ( 13( 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 ( (2)(