 2.1.1: Let A=[ ], B=[ ! 9 3 5 C = [ !]. and D = [ ] ] . 7 Compute the...
 2.1.2: Let A= [ ]. B = [ = ]. 1 4 5 9 C= H!  nandD= nl Compute the foll...
 2.1.3: Let A = [ ]. B = [ _ ]. C = [J.andD = [ ] Compute the following (i...
 2.1.4: LdA = nlB = [ 1 n 7 3 c = [2 0 S],andD = [ : 4  ] Compute the f...
 2.1.5: Let A= [ 1] [ 1 ,B = C= [ 4 3 ] .and D = [  n ] Compute the foll...
 2.1.6: LetA= 5 6 2 0 ;],andB = [ 1 1 2 3 ] 6 7 . 0 4 Let 03 and /3 be ...
 2.1.7: a) Let A be an n X n matrix and X be an n X 1 column matrix of 1 s....
 2.1.8: LetA be a 3 X 5 matrix, Ba 5 X 2 matrix, Ca 3 X 4 matrix, D a 4 X 2...
 2.1.9: LetA be a 2 X 2 matrix, Ba 2 X 2 matrix, Ca 2 X 3 matrix, D a 3 X 2...
 2.1.10: Let C = AB and D = BA for the following matrices A and B. 3 5] [ ...
 2.1.11: Let R = PQ and S = QP, where P = [ i and Q = [  1 3 3] 4 Determi...
 2.1.12: If A = O 4 , B = 5 O  l , and C = [  ] determine the following el...
 2.1.13: If A B= [ 0 1 1 2] 0 4 , and 3 2 0 2] 7 5 , determine the follo...
 2.1.14: Let A = 3 O , B = 4 l , C = 4 2 0 ] Compute the following products...
 2.1.15: Let A [: 2 !J. B [j} 2 5 p = [ 0 2 ] . Q 6 7 [] (a) Express the ...
 2.1.16: Let A and B be the following matrices. Compute row 2 of the matrix ...
 2.1.17: Let A be a matrix whose third row is all zeros. Let Bbe any matrix ...
 2.1.18: Let D be a matrix whose second column is all zeros. Let C be any ma...
 2.1.19: LetA be an m X r matrix, Ban r X n matrix, and C = AB. Let the colu...
 2.1.20: Let A and B be the following matrices. Use the result of Exercise 1...
 2.1.21: Use the given partitions of A and B below to compute AB. (a) A [ + ...
 2.1.22: Let A [1 2] andB = . For each partition of A given below find all...
 2.1.23: UtA [  ] mdB l  Jl For each partition of B given below find all ...
 2.1.24: Suggest suitable partitions involving zero and identity submatrices...
 2.1.25: State (with a brief explanation) whether the following statements a...
Solutions for Chapter 2.1: Addition, Scalar Multiplication, and Multiplication of Matrices
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 2.1: Addition, Scalar Multiplication, and Multiplication of Matrices
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.1: Addition, Scalar Multiplication, and Multiplication of Matrices includes 25 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Since 25 problems in chapter 2.1: Addition, Scalar Multiplication, and Multiplication of Matrices have been answered, more than 9538 students have viewed full stepbystep solutions from this chapter.

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

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.)

Column space C (A) =
space of all combinations of the columns of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Outer product uv T
= column times row = rank one matrix.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

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.