 3.2.1: In Exercises 16, perform the indicated computations whenpossible, u...
 3.2.2: In Exercises 16, perform the indicated computations whenpossible, u...
 3.2.3: In Exercises 16, perform the indicated computations whenpossible, u...
 3.2.4: In Exercises 16, perform the indicated computations whenpossible, u...
 3.2.5: In Exercises 16, perform the indicated computations whenpossible, u...
 3.2.6: In Exercises 16, perform the indicated computations whenpossible, u...
 3.2.7: In Exercises 710, find the missing values in the given matrixequati...
 3.2.8: In Exercises 710, find the missing values in the given matrixequati...
 3.2.9: In Exercises 710, find the missing values in the given matrixequati...
 3.2.10: In Exercises 710, find the missing values in the given matrixequati...
 3.2.11: Find all values of a such that A2 = A forA =5 10a 4
 3.2.12: Find all values of a such that A3 = 2A forA =2 21 a
 3.2.13: Let T1 and T2 be linear transformations given byT1x1x2 = 3x1 + 5x22...
 3.2.14: Let T1 and T2 be linear transformations given byT1x1x2 =2x1 + 3x2x1...
 3.2.15: In Exercises 1518, expand each of the given matrix expressionsand c...
 3.2.16: In Exercises 1518, expand each of the given matrix expressionsand c...
 3.2.17: In Exercises 1518, expand each of the given matrix expressionsand c...
 3.2.18: In Exercises 1518, expand each of the given matrix expressionsand c...
 3.2.19: In Exercises 1922, the given matrix equation is nottrue in general....
 3.2.20: In Exercises 1922, the given matrix equation is nottrue in general....
 3.2.21: In Exercises 1922, the given matrix equation is nottrue in general....
 3.2.22: In Exercises 1922, the given matrix equation is nottrue in general....
 3.2.23: Suppose that A has four rows and B has five columns. If ABis define...
 3.2.24: Suppose that A has four rows and B has five columns. If B Ais defin...
 3.2.25: In Exercises 2528,A =1 2 1 32 0 141 2 2 00 1 21, B =2 0 1 131210 1 ...
 3.2.26: In Exercises 2528,A =1 2 1 32 0 141 2 2 00 1 21, B =2 0 1 131210 1 ...
 3.2.27: In Exercises 2528,A =1 2 1 32 0 141 2 2 00 1 21, B =2 0 1 131210 1 ...
 3.2.28: In Exercises 2528,A =1 2 1 32 0 141 2 2 00 1 21, B =2 0 1 131210 1 ...
 3.2.29: Suppose that A is a 3 3 matrix. Find a 3 3 matrix E suchthat the pr...
 3.2.30: Suppose that A is a 4 3 matrix. Find a 4 4 matrix E suchthat the pr...
 3.2.31: FIND AN EXAMPLE For Exercises 3138, find an example thatmeets the g...
 3.2.32: FIND AN EXAMPLE For Exercises 3138, find an example thatmeets the g...
 3.2.33: FIND AN EXAMPLE For Exercises 3138, find an example thatmeets the g...
 3.2.34: FIND AN EXAMPLE For Exercises 3138, find an example thatmeets the g...
 3.2.35: FIND AN EXAMPLE For Exercises 3138, find an example thatmeets the g...
 3.2.36: FIND AN EXAMPLE For Exercises 3138, find an example thatmeets the g...
 3.2.37: FIND AN EXAMPLE For Exercises 3138, find an example thatmeets the g...
 3.2.38: FIND AN EXAMPLE For Exercises 3138, find an example thatmeets the g...
 3.2.39: TRUE OR FALSE For Exercises 3948, determine if the statementis true...
 3.2.40: TRUE OR FALSE For Exercises 3948, determine if the statementis true...
 3.2.41: TRUE OR FALSE For Exercises 3948, determine if the statementis true...
 3.2.42: TRUE OR FALSE For Exercises 3948, determine if the statementis true...
 3.2.43: TRUE OR FALSE For Exercises 3948, determine if the statementis true...
 3.2.44: TRUE OR FALSE For Exercises 3948, determine if the statementis true...
 3.2.45: TRUE OR FALSE For Exercises 3948, determine if the statementis true...
 3.2.46: TRUE OR FALSE For Exercises 3948, determine if the statementis true...
 3.2.47: TRUE OR FALSE For Exercises 3948, determine if the statementis true...
 3.2.48: TRUE OR FALSE For Exercises 3948, determine if the statementis true...
 3.2.49: Prove the remaining unproven parts of Theorem 3.11.(a) A + B = B + ...
 3.2.50: Prove the remaining unproven parts of Theorem 3.13.(a) A(BC) = (AB)...
 3.2.51: Prove the remaining unproven parts of Theorem 3.15.(a) (A + B)T = A...
 3.2.52: Verify Equation (2): If A is an nm matrix and In is the nnidentity ...
 3.2.53: Show that if A and B are symmetric matrices and AB = B A,then AB is...
 3.2.54: Let A and D be n n matrices, and suppose that the onlynonzero terms...
 3.2.55: Let A be an n m matrix.(a) What are the dimensions of AT A?(b) Show...
 3.2.56: Suppose that A and B are both n n diagonal matrices. Provethat AB i...
 3.2.57: Suppose that A and B are both n n upper triangular matrices.Prove t...
 3.2.58: Suppose that Aand B are both nn lower triangular matrices.Prove tha...
 3.2.59: Prove Theorem 3.17: If Ais an upper (lower) triangular matrixand k ...
 3.2.60: If A is a square matrix, show that A + AT is symmetric.
 3.2.61: A square matrix A is skew symmetric if AT = A.(a) Find a 3 3 skew s...
 3.2.62: A square matrix A is idempotent if A2 = A.(a) Find a 2 2 matrix, no...
 3.2.63: If A is a square matrix, show that (AT )T = A.
 3.2.64: The trace of a square matrix A is the sum of the diagonal termsof A...
 3.2.65: C In Example 7, suppose that the current distribution is 8000homes ...
 3.2.66: C In Example 7, suppose that the current distribution is 5000homes ...
 3.2.67: C In an office complex of 1000 employees, on any given daysome are ...
 3.2.68: C The star quarterback of a university football team has decidedto ...
 3.2.69: C In Exercises 6974, perform the indicated computations whenpossibl...
 3.2.70: C In Exercises 6974, perform the indicated computations whenpossibl...
 3.2.71: C In Exercises 6974, perform the indicated computations whenpossibl...
 3.2.72: C In Exercises 6974, perform the indicated computations whenpossibl...
 3.2.73: C In Exercises 6974, perform the indicated computations whenpossibl...
 3.2.74: C In Exercises 6974, perform the indicated computations whenpossibl...
Solutions for Chapter 3.2: Matrix Algebra
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 3.2: Matrix Algebra
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Since 74 problems in chapter 3.2: Matrix Algebra have been answered, more than 15023 students have viewed full stepbystep solutions from this chapter. Chapter 3.2: Matrix Algebra includes 74 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Iterative method.
A sequence of steps intended to approach the desired solution.

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

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).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.