 2.1.1E: [M] Let An be the n × n matrix with 0’s on the main diagonal and 1’...
 2.1.2E: In Exercises, compute each matrix sum or product if it is defined. ...
 2.1.3E: In this exercise, you should assume that each matrix expression is ...
 2.1.4E: Compute A ? 5I3 and (5I3)A, when
 2.1.5E: In Exercises, compute the product AB in two ways:(a) by the definit...
 2.1.6E: In Exercises, compute the product AB in two ways:(a) by the definit...
 2.1.7E: If a matrix A is 5 × 3 and the product AB is 5 × 7, what is the siz...
 2.1.8E: How many rows does B have if BC is a 3 × 4 matrix?
 2.1.9E: Let and . What value(s) of k, if any, will make AB = BA?
 2.1.10E: Let , , and . Verify that AB = AC and yet B ? C.
 2.1.11E: Let and . Compute AD and DA. Explain how the columns or rows of A c...
 2.1.12E: Construct a 2 × 2 matrix B such that AB is the zero matrix. Use two...
 2.1.13E: Let r1,…, rp be vectors in ?n, and let Q be an m × n matrix. Write ...
 2.1.14E: Let U be the 3 × 2 cost matrix described in Example 6 in Section 1....
 2.1.15E: Exercises 15 and 16 concern arbitrary matrices A, B, and C for whic...
 2.1.16E: Exercises concern arbitrary matrices A, B, and C for which the indi...
 2.1.17E: If and , determine the first and second columns of B.
 2.1.18E: Suppose the first two columns, b1 and b2, of B are equal. What can ...
 2.1.19E: Suppose the third column of B is the sum of the first two columns. ...
 2.1.20E: Suppose the second column of B is all zeros. What can you say about...
 2.1.21E: Suppose the last column of AB is entirely zeros but B itself has no...
 2.1.22E: Show that if the columns of B are linearly dependent, then so are t...
 2.1.23E: Suppose CA = In (the n × n identity matrix). Show that the equation...
 2.1.24E: Suppose AD = Im (the m × m identity matrix). Show that for any b in...
 2.1.25E: Suppose A is an m × n matrix and there exist n × m matrices C and D...
 2.1.26E: Suppose A is a 3 × n matrix whose columns span ?3. Explain how to c...
 2.1.27E: In Exercises, view vectors in ?n as n × 1 matrices. For u and v in ...
 2.1.28E: In Exercises 27 and 28, view vectors in T : ?n as n × 1 matrices. F...
 2.1.29E: Prove Theorem 2(b) and 2(c). Use the row–column rule. The (I, j)en...
 2.1.30E: Prove Theorem 2(d). [Hint: The (I, j) entry in (r A)B is Theorem 2(...
 2.1.31E: Show that Im A = A where A is an m × n matrix. Assume Imx = x for a...
 2.1.32E: Show that AIn= A when A is an m × n matrix. [Hint: Use the (column)...
 2.1.33E: Prove Theorem 3(d). [Hint: Consider the j th row of (AB)T]Theorem 3...
 2.1.34E: Give a formula for (ABx)T , where x is a vector and A and B are mat...
 2.1.35E: [M] Read the documentation for your matrix program, and write the c...
 2.1.36E: [M] Write the command(s) that will create a 6 × 4 matrix with rando...
 2.1.37E: [M] Construct a random 5 × 5 matrix A and test whether The best way...
 2.1.38E: [M] Use at least three pairs of random 4 × 4 matrices A and B to te...
 2.1.39E: [M] Let Compute Sk for k = 2,…, 6.
 2.1.40E: [M] Describe in words what happens when you compute A5, A10, A20, a...
 2.1.41E: [M] Describe in words what happens when you compute A5, A10, A20, a...
Solutions for Chapter 2.1: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 2.1
Get Full SolutionsLinear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Chapter 2.1 includes 41 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 2.1 have been answered, more than 48086 students have viewed full stepbystep solutions from this chapter.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

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

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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

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.

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

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

Solvable system Ax = b.
The right side b is in the column space of A.

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.