 2.2.1E: Find the inverses of the matrices in Exercises 1–4.
 2.2.2E: Find the inverses of the matrices in Exercises 1–4.
 2.2.3E: Find the inverses of the matrices in Exercises 1–4.
 2.2.4E: Find the inverses of the matrices in Exercises 1–4.
 2.2.5E: Use the inverse found in Exercise 1 to solve the systemExercise 1:F...
 2.2.6E: Use the inverse found in Exercise 3 to solve the systemExercise 3:F...
 2.2.7E: a. Find A–1 and use it to solve the four equationsb. The four equat...
 2.2.8E: Suppose P is invertible and A = PBP–1 Solve for B in terms of A.
 2.2.9E: In Exercises 9 and 10, mark each statement True or False. Justify e...
 2.2.10E: In Exercises 9 and 10, mark each statement True or False. Justify e...
 2.2.11E: Let A be an invertible n × n matrix, and let B be an n × p matrix. ...
 2.2.12E: Use matrix algebra to show that if A is invertible and D satisfies ...
 2.2.13E: Suppose AB = AC, where B and C are n × p matrices and A is invertib...
 2.2.14E: Suppose (B – C)D = 0 where B and C are m × n matrices and D is inve...
 2.2.15E: Let A be an invertible n × n matrix, and let B be an n × p matrix. ...
 2.2.16E: Suppose A and B are n × n matrices, B is invertible, and AB is inve...
 2.2.17E: Suppose A, B, and C are invertible n × n matrices. Show that ABC is...
 2.2.18E: Solve the equation AB = BC for A, assuming that A, B, and C are squ...
 2.2.19E: If A, B and C are n × n invertible matrices, does the equationhave ...
 2.2.20E: Suppose A, B, and X are n × n matrices with A, X, and A – AX invert...
 2.2.21E: Explain why the columns of an n × n matrix A are linearly independe...
 2.2.22E: Explain why the columns of an n × n matrix A span ?n when A is inve...
 2.2.23E: Suppose A is n × n and the equation Ax = 0 has only the trivial sol...
 2.2.24E: Suppose A is n × n and the equation Ax = b has a solution for each ...
 2.2.25E: Exercises 25 and 26 prove Theorem 4 for Show that if ad – bc = 0, t...
 2.2.26E: Exercises 25 and 26 prove Theorem 4 for Show that if ad – bc ? 0, t...
 2.2.27E: Exercises 27 and 28 prove special cases of the facts about elementa...
 2.2.28E: Exercises 27 and 28 prove special cases of the facts about elementa...
 2.2.29E: Find the inverses of the matrices in Exercises 29–32, if they exist...
 2.2.30E: Find the inverses of the matrices in Exercises 29–32, if they exist...
 2.2.31E: Find the inverses of the matrices in Exercises 29–32, if they exist...
 2.2.32E: Find the inverses of the matrices in Exercises 29–32, if they exist...
 2.2.33E: Use the algorithm from this section to find the inverses of A be th...
 2.2.34E: Repeat the strategy of Exercise 33 to guess the inverse B of Show t...
 2.2.35E: Let Find the third column of A–1 without computing the other columns.
 2.2.36E: [M] Let Find the second and third columns of A–1 without computing ...
 2.2.37E: Let Construct a 2 × 3 matrix C (by trial and error) using only 1, –...
 2.2.38E: Let Construct a 4 ? 2 matrix D using only 1 and 0 as entries, such ...
 2.2.39E: [M] Let be a flexibility matrix, with flexibility measured in inche...
 2.2.40E: [M] Compute the stiffness matrix D–1 for D in Exercise 39. List the...
 2.2.41E: [M] Let be a flexibility matrix for an elastic beam such as the one...
 2.2.42E: [M] With D as in Exercise 41, determine the forces that produce a d...
Solutions for Chapter 2.2: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 2.2
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. Chapter 2.2 includes 42 full stepbystep solutions. Since 42 problems in chapter 2.2 have been answered, more than 34725 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

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!).

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.