 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 4335 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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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