 1.4.1: Explain why each of the following algebraic rules will not work in ...
 1.4.2: Will the rules in Exercise 1 work if a is replaced by an n n matrix...
 1.4.3: Find nonzero 2 2 matrices A and B such that AB = O.
 1.4.4: Find nonzero matrices A, B, and C such that AC = BC and A _= B
 1.4.5: The matrix A = 1 1 1 1 has the property that A2 = O. Is it possible...
 1.4.6: Prove the associative law of multiplication for 22 matrices; that i...
 1.4.7: Let A = 1 2 1 2 1 2 1 2 Compute A2 and A3 . What will An turn out t...
 1.4.8: Let A = 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2...
 1.4.9: Let A = 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 Show that An = O for n 4. 1
 1.4.10: Let A and B be symmetric n n matrices. For each of the following, d...
 1.4.11: Let C be a nonsymmetric n n matrix. For each of the following, dete...
 1.4.12: Let A = a11 a12 a21 a22 Show that if d = a11a22 a21a12 _= 0, then A...
 1.4.13: Use the result from Exercise 12 to find the inverse of each of the ...
 1.4.14: Let A and B be n n matrices. Show that if AB = A and B _= I then A ...
 1.4.15: Let A be a nonsingular matrix. Show that A1 is also nonsingular and...
 1.4.16: Prove that if A is nonsingular, then AT is nonsingular and (AT ) 1 ...
 1.4.17: Let A be an n n matrix and let x and y be vectors in Rn. Show that ...
 1.4.18: Let A be a nonsingular n n matrix. Use mathematical induction to pr...
 1.4.19: Let A be an n n matrix. Show that if A2 = O, then I A is nonsingula...
 1.4.20: Let A be an n n matrix. Show that if Ak+1 = O, then I A is nonsingu...
 1.4.21: Given R = cos sin sin cos show that R is nonsingular and R1 = RT . 2
 1.4.22: An n n matrix A is said to be an involution if A2 = I . Show that i...
 1.4.23: Let u be a unit vector in Rn (i.e., uT u = 1) and let H = I 2uuT . ...
 1.4.24: A matrix A is said to be idempotent if A2 = A. Show that each of th...
 1.4.25: Let A be an idempotent matrix. (a) Show that I A is also idempotent...
 1.4.26: Let D be an n n diagonal matrix whose diagonal entries are either 0...
 1.4.27: Let A be an involution matrix, and let B = 1 2 (I + A) and C = 1 2 ...
 1.4.28: Let A be an mn matrix. Show that ATA and AAT are both symmetric. 2
 1.4.29: Let A and B be symmetric n n matrices. Prove that AB = BA if and on...
 1.4.30: Let A be an n n matrix and let B = A + AT and C = A AT (a) Show tha...
 1.4.31: In Application 1, how many married women and how many single women ...
 1.4.32: Consider the matrix A = 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 0 0 1 1 0...
 1.4.33: Consider the graph V2 V3 V5 V1 V4 (a) Determine the adjacency matri...
 1.4.34: If Ax = Bx for some nonzero vector x, then the matrices A and B mus...
 1.4.35: If A and B are singular n n matrices, then A+ B is also singular. 3
 1.4.36: If A and B are nonsingular matrices, then (AB)T is nonsingular and ...
Solutions for Chapter 1.4: Matrix Algebra
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 1.4: Matrix Algebra
Get Full SolutionsSince 36 problems in chapter 1.4: Matrix Algebra have been answered, more than 6801 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.4: Matrix Algebra includes 36 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.