 1.5.1: Which of the matrices that follow are elementary matrices? Classify...
 1.5.2: Find the inverse of each matrix in Exercise 1. For each elementary ...
 1.5.3: For each of the following pairs of matrices, find an elementary mat...
 1.5.4: For each of the following pairs of matrices, find an elementary mat...
 1.5.5: Let A = 1 2 4 2 1 3 1 0 2 , B = 1 2 4 2 1 3 2 2 6 , C = 1 2 4 0 1 3...
 1.5.6: Let A = 2 1 1 6 4 5 4 1 3 (a) Find elementary matrices E1, E2, E3 s...
 1.5.7: Let A = 2 1 6 4 (a) Express A as a product of elementary matrices. ...
 1.5.8: Compute the LU factorization of each of the following matrices: (a)...
 1.5.9: Let A = 1 0 1 3 3 4 2 2 3 (a) Verify that A1 = 1 2 3 1 1 1 0 2 3 (b...
 1.5.10: Find the inverse of each of the following matrices: (a) 1 1 1 0 (b)...
 1.5.11: Given A = 3 1 5 2 and B = 1 2 3 4 compute A1 and use it to (a) find...
 1.5.12: Let A = 5 3 3 2 , B = 6 2 2 4 ,C = 4 2 6 3 Solve each of the follow...
 1.5.13: Is the transpose of an elementary matrix an elementary matrix of th...
 1.5.14: Let U and R be nn upper triangular matrices and set T = UR. Show th...
 1.5.15: Let A be a 3 3 matrix and suppose that 2a1 + a2 4a3 = 0 How many so...
 1.5.16: Let A be a 3 3 matrix and suppose that a1 = 3a2 2a3 Will the system...
 1.5.17: Let A and B be n n matrices and let C = A B. Show that if Ax0 = Bx0...
 1.5.18: Let A and B be n n matrices and let C = AB. Prove that if B is sing...
 1.5.19: Let U be an n n upper triangular matrix with nonzero diagonal entri...
 1.5.20: Let A be a nonsingular nn matrix and let B be an n r matrix. Show t...
 1.5.21: In general, matrix multiplication is not commutative (i.e., AB _= B...
 1.5.22: Show that if A is a symmetric nonsingular matrix, then A1 is also s...
 1.5.23: Prove that if A is row equivalent to B, then B is row equivalent to...
 1.5.24: (a) Prove that if A is row equivalent to B and B is row equivalent ...
 1.5.25: Let A and B be m n matrices. Prove that if B is row equivalent to A...
 1.5.26: Prove that B is row equivalent to A if and only if there exists a n...
 1.5.27: Is it possible for a singular matrix B to be row equivalent to a no...
 1.5.28: Given a vector x Rn+1, the (n + 1) (n + 1) matrix V defined by vi j...
 1.5.29: If A is row equivalent to I and AB = AC, then B must equal C. 3
 1.5.30: If E and F are elementary matrices and G = EF, then G is nonsingula...
 1.5.31: If A is a 4 4 matrix and a1 +a2 = a3 +2a4, then A must be singular. 3
 1.5.32: If A is row equivalent to both B and C, then A is row equivalent to...
Solutions for Chapter 1.5: Elementary Matrices
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 1.5: Elementary Matrices
Get Full SolutionsSince 32 problems in chapter 1.5: Elementary Matrices have been answered, more than 4186 students have viewed full stepbystep solutions from this chapter. Chapter 1.5: Elementary Matrices includes 32 full stepbystep solutions. 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. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290.

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!

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.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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).

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.