 3.2.1: In Exercises I I 0. evaluate the determinant of each matrix using ...
 3.2.2: In Exercises I I 0. evaluate the determinant of each matrix using ...
 3.2.3: In Exercises I I 0. evaluate the determinant of each matrix using ...
 3.2.4: In Exercises I I 0. evaluate the determinant of each matrix using ...
 3.2.5: In Exercises I I 0. evaluate the determinant of each matrix using ...
 3.2.6: In Exercises I I 0. evaluate the determinant of each matrix using ...
 3.2.7: In Exercises I I 0. evaluate the determinant of each matrix using ...
 3.2.8: In Exercises I I 0. evaluate the determinant of each matrix using ...
 3.2.9: In Exercises I I 0. evaluate the determinant of each matrix using ...
 3.2.10: In Exercises I I 0. evaluate the determinant of each matrix using ...
 3.2.11: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.12: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.13: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.14: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.15: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.16: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.17: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.18: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.19: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.20: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.21: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.22: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.23: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.24: In Exercises I 1 24, evaluate the dererminam of each matrix using ...
 3.2.25: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.26: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.27: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.28: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.29: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.30: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.31: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.32: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.33: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.34: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.35: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.36: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.37: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.38: For each of tile mmrices in rerr:ises 25 38, determine tile value(...
 3.2.39: The determinant of a square matrix equals the product of its diagon...
 3.2.40: Performing a row addition operation on a square matrix does not cha...
 3.2.41: Performing a >ealing operation on a quare matrix does not change it...
 3.2.42: Performing an interchange operation on a square matrix changes its ...
 3.2.43: For any 11 x 11 matrices A ~md 8, we have del (A+ 8 ) = dctA + detB.
 3.2.44: For any 11 x 11 matrices A and B. dctA8 = (detA)(detB).
 3.2.45: If A is any invenible matrix. then detA = 0.
 3.2.46: For any square matrix A, detA 1 =  detA.
 3.2.47: The determinant of any square matrix c;m be evaluated by a cofactor...
 3.2.48: The determinant of any square matrix equals the product of the diag...
 3.2.49: If detA :f= 0. then A is an invenible matrix.
 3.2.50: The detemtinant of the 11 x 11 identity matrix is l.
 3.2.51: If A i any square matrix ;md I ' b any scalar. then dct cA = cdetA.
 3.2.52: Cramer's mle can be used to solve :my syMem of 11 linear equations ...
 3.2.53: To solve a system of 5 linear equations in 5 ' 'ariables with Cmmer...
 3.2.54: f A is an invertible matrix, then det A
 3.2.55: If A is a 4 x 4 matrix. then del ( A)= dctA.
 3.2.56: f A is a 5 x 5 matrix. then del ( A) =
 3.2.57: For any square matrix A and any po:,iti\'e integer k. det (At )= (d...
 3.2.58: If an 11 x 11 matrix A is transformed into an upper triangular matr...
 3.2.59: /11 E.tucius 5966, solle each sysrem11sing Cmmer's rule. 59. ' + 2...
 3.2.60: /11 E.tucius 5966, solle each sysrem11sing Cmmer's rule. 60. 2tt+3...
 3.2.61: /11 E.tucius 5966, solle each sysrem11sing Cmmer's rule. 61. lit +...
 3.2.62: /11 E.tucius 5966, solle each sysrem11sing Cmmer's rule. 62. l rt ...
 3.2.63: /11 E.tucius 5966, solle each sysrem11sing Cmmer's rule. 63. xl + ...
 3.2.64: /11 E.tucius 5966, solle each sysrem11sing Cmmer's rule. 64. ' 2 +...
 3.2.65: /11 E.tucius 5966, solle each sysrem11sing Cmmer's rule. 65. XI Xj...
 3.2.66: /11 E.tucius 5966, solle each sysrem11sing Cmmer's rule. 66. 1rt +...
 3.2.67: Give an example to show that detkA :f= k detA for some matrix A and...
 3.2.68: E\aluate dct kA if A is an 11 x 11 matnx and k is a scalar. Ju,tify...
 3.2.69: Pro~e that if A is :m in"enible mmnx. then dctA  I = I detA
 3.2.70: Under what conditions is det (A)=  detA? JuMify )Our :tns\\er.
 3.2.71: Let A and 8 be 11 x 11 matrices ~uch that 8 i' in\'ertible. Prove t...
 3.2.72: An 11 x 11 m;urix A is called 11ilpore111 if. for some positive int...
 3.2.73: An 11 x 11 matrix Q is called orrhogorra/ if Q r Q = 1,. Prove that...
 3.2.74: A square matrix A is called skew.rymmerric if AT =  A. Prove that ...
 3.2.75: The matrix A = [: a b
 3.2.76: Use pi'Openies of determinants to show that the equation of the lin...
 3.2.77: l.ct 8 = fbt. b2 .. .. . bnl be a subset of 'R.n containing 11 dist...
 3.2.78: Let A be an 11 x 11 matrix with rows a' 1, a; , .. .. a;, and B be ...
 3.2.79: Complete the proof of Theorem 3.3(b).
 3.2.80: Complete the proof of Theorem 3.3(d).
 3.2.81: Prove that det T = dct for every elementary matrix E.
 3.2.82: l.ct A be an 11 x 11 matrix and b1 , denote the (k .j)cofactor of A...
 3.2.83: In &ercises 83 85. IISI' ei1her ft c:alclllmor ll'illt mlllrix cap...
 3.2.84: In &ercises 83 85. IISI' ei1her ft c:alclllmor ll'illt mlllrix cap...
 3.2.85: In &ercises 83 85. IISI' ei1her ft c:alclllmor ll'illt mlllrix cap...
Solutions for Chapter 3.2: DETERMINANTS
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Solutions for Chapter 3.2: DETERMINANTS
Get Full SolutionsElementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. Since 85 problems in chapter 3.2: DETERMINANTS have been answered, more than 34899 students have viewed full stepbystep solutions from this chapter. Chapter 3.2: DETERMINANTS includes 85 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.