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1、线性代数习题解答赵树嫄（Linear algebra answers Zhao Shuyuan）Exercises for Linear Algebra (Zhou 3)Instructions: 1. Teaching materials Zhao Shuyuan. Linear Algebra (Fourth Edition), Renmin University of China press, 19972, the answer is only for reference; if there is an error, look back.Problem one (A)1, (6)(7)2
2、, 7 (3)(4) 04, or58, (1) 4 (2), 7 (3) 13(4) N (n (n-1). 21) = (n-1) + (n-2) +. +2+1=10, the column number is 3k42l, so K and l can choose 1 or 5; if k=1, l=5, N (31425) =3, the k=1, l=5. is negative;12, (1) an item not equal to zero(2)!13, (3)(4) add the columns to the first column,17, (1) start wit
3、h the second line, with each line plus the first line.(2).(3) the sum of the columns is equal, and each row is added to the first row.18, (3)20, the first line is added to each line to get the upper triangle determinant,21, the sum of the rows is equal, the columns are added to the first column, and
4、 the common factor is put forwardStarting from the second line, subtracting the first row from each row22, the last column is multiplied by first and 2, respectively,. The N-1 column gets the upper triangle determinant23, expand in the first column24, the first column second column third column seco
5、nd column, then,. Column n, column n-1, and finally expand in line 1.25, (1)(2) the sum of the lines is equal.(3) similar to the 22 question.(4) at that time, substituting the determinant would make the determinant have two lines of the same, so they were all the root of the equation.28,29, in which
6、 1, 3, two lines correspond to proportion, so it is zero32, start from the second row of each row is multiplied by (- 1) on the line to the first column and then press start33, expand in the first column34, the primitive equation is transformed into. .35,= 0Solution or36, (Vandermonde determinant)37
7、, Xie40, (3) D=63, D1=63, D2=126, D3=189(6) D=20, D1=60, D2=-80, D3=-20, D4=2042 dreams.Hence only the zero solution of the original equation.43 order,When or when the system of homogeneous equations has nonzero solutions.44, the coefficient determinant of the homogeneous system of equationsThat is,
8、 the system of homogeneous equations has only zero solutions at the same time.Problem two (A)2, (1)(2)(3)(4) 3Y=2 (A+B) is obtained by (2A - Y) +2 (B - Y) =0L3, because the equations are obtainedThe solutions are x = = 5, y = = 6, u = 4, v = - 25, (2)(3) 14(7)11, (1) set, thenEquations are obtainedX
9、ie jie,And solution.(2)(3) set up,The solution is so13. set all exchangeable matrices as follows,The solution thus comes16, (3) because, so(4) because it can be pushed by mathematical induction.(5) because it can be introduced.20,21.28, because,So its symmetric matrixBecause, so it is symmetric matr
10、ix31, (1) the original matrix for which;(3) the original matrix is as follows.33,34, (2) because, so(4) because it is reversible(6) because it is reversible,.40, (1)(2)(3).42, by get,.44, multiply both sides45 by obtaining, then reversible, and.51, because,.52.53, (3) elementary rows are obtained by
11、 transformation(6),.54, (1),So(4),.55, (1),.(2),.56,.57, (1), rank 2.(3)Rank 3.(4) rank 3.58, the elementary row transformation is obtained because the rank of 2 must be,.59,Dangdang60,Because, therefore, second, 32 lines in proportion to obtainXie jie,Problem three (A)1, the following linear equati
12、ons are solved by the elimination method(1)solutionHui hui,The system of equations has unique solutions:(2)Solution:,The rank of the coefficient matrix is 2, while the rank of the augmented matrix is 3; the equations are free of solution(3)Solution: (A, b) =The equations of the same solution are obt
13、ainedSuppose that the general solution is(6)Solution: A =The equations of the same solution are obtainedOrder,obtain2, determine the values of a and B so that the following linear equations are solvable and their solutions are obtained(2)Solution: the coefficient determinant of equation D=When the s
14、um of time, the equation has unique solution,HenceAt that time, the equations were set, and the equations were infinitely many solutions,;At that time, the equations were as follows, and their augmented matrices were(A, b) = = R (A) =2, R (A, b) =3, and equations are notSolution.Supplement,Solution:
15、At this point, the augmented matrix isInterpret as;When there are infinitely many solutions,When there are infinitely many solutions,There are infinitely many solutions,3, (1)(2)4, (1),(2)6, (1) (a) set,have toReduced to equations,L(b) elementary row transformation of a matrix:Available(2)9, set by
16、questionR =.Namely,10, (1) the matrix is knowableLinear correlation(2) the matrix is linearly independent11, the determinant of a matrix consisting of a corresponding vector is equal toLinearly independent12, a matrix consisting of corresponding vectors,Dreams, so, the linear correlation.13, to prov
17、e: order,Tidy upBecause the line is independent, so there isThe solution vector is linearly independent14 order,At that time, linearly independent; then linear correlation16, (1) the elementary transformation of the matrix is obtained,L is the maximum linearly independent group, -(2) the elementary
18、transformation of the matrix is obtainedMaximum linear independent group,17, the primary row transformation is applied(1)Hence, is the maximum linearly independent group; and,(2)Is a maximal linearly independent group; and,20, (1) transform the coefficient matrixEquation setThe order is the basis of
19、 the solution(2)Equation setOrder to get:And then we get the basic solution(3)Equations are obtainedThe fundamental solution is obtained23, the coefficients or augmented matrices are transformed(1) equations are obtainedOrder to getThe fundamental solution is, where C is any constant(2)Equation set,
20、The corresponding homogeneous linear equations areSo, special solution,Order again,The fundamental solution isThe general solution of the original equation set is, where it is any constant(3),Equations are obtainedParticular solutions, fundamental solutions,So the whole solution is24,The discussion
21、follows:(1) there was no solution to the equations at that time;(2) there was only one solution at that time;(3) there were infinitely many solutions at that timeThe basic solution isFor particular solution, all solutions are.25, the augmented matrix is transformed into a T matrixKnowableWhen and on
22、ly = 0, the equations have solutions; the general solution is(that is, any real number)Problem four (A)1, (1) by getting the characteristic value as,.(2) from= 0,That is,.(3)= 0The characteristic value isBy substituting.Well,.(4),Get,When the fundamental solution system is obtained, all the correspo
23、nding eigenvectors are(not all zeros),At that time, the fundamental equations were obtained by solving the system of equationsThe whole feature vector is3, from the question setting,(1) that is, the eigenvalues are(2) reversible by A,The eigenvalues of(3)The eigenvalues of4, set up,5, in order to substituteGetSubstitutionXie Jie.So other eigenvalues are8, if A is reversible, it exists, andStar.?