 7.2.1: If A =1 2 03 2 1213and B =4 2 3150612, find(a) 2A + B (b) A 4B(c) A...
 7.2.2: If A =1 + i 1 + 2i3 + 2i 2 iand B =i 32 2i, find(a) A 2B (b) 3A + B...
 7.2.3: If A =2121 0 32 1 1and B =1 2 33 1 1210, find(a) AT (b) BT(c) AT + ...
 7.2.4: If A =3 2i 1 + i2 i 2 + 3i, find(a) AT (b) A (c) A
 7.2.5: If A =3 2 12 1 2121and B =2 1 1233102, verify that 2(A + B) = 2A + 2B.
 7.2.6: If A =1 2 03 2 1203, B =2 1 1233102, and C =2101220 1 1, verify tha...
 7.2.7: Prove each of the following laws of matrix algebra:(a) A + B = B + ...
 7.2.8: If x =23i1 iand y =1 + i23 i, find(a) xT y (b) yT y(c) (x, y) (d) (...
 7.2.9: If x =1 2ii2and y =23 i1 + 2i, show that(a) xT y = yT x (b) (x,y) =...
 7.2.10: In each of 10 through 19, either compute the inverse of the given m...
 7.2.11: In each of 10 through 19, either compute the inverse of the given m...
 7.2.12: In each of 10 through 19, either compute the inverse of the given m...
 7.2.13: In each of 10 through 19, either compute the inverse of the given m...
 7.2.14: In each of 10 through 19, either compute the inverse of the given m...
 7.2.15: In each of 10 through 19, either compute the inverse of the given m...
 7.2.16: In each of 10 through 19, either compute the inverse of the given m...
 7.2.17: In each of 10 through 19, either compute the inverse of the given m...
 7.2.18: In each of 10 through 19, either compute the inverse of the given m...
 7.2.19: In each of 10 through 19, either compute the inverse of the given m...
 7.2.20: If A is a square matrix, and if there are two matrices B and C such...
 7.2.21: If A(t) =et 2et e2t2et et e2tet 3et 2e2tand B(t) =2et et 3e2tet 2et...
 7.2.22: In each of 22 through 24, verify that the given vector satisfies th...
 7.2.23: In each of 22 through 24, verify that the given vector satisfies th...
 7.2.24: In each of 22 through 24, verify that the given vector satisfies th...
 7.2.25: In each of 25 and 26, verify that the given matrix satisfies the gi...
 7.2.26: In each of 25 and 26, verify that the given matrix satisfies the gi...
Solutions for Chapter 7.2: Review of Matrices
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 7.2: Review of Matrices
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Since 26 problems in chapter 7.2: Review of Matrices have been answered, more than 11656 students have viewed full stepbystep solutions from this chapter. Chapter 7.2: Review of Matrices includes 26 full stepbystep solutions. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

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.

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.

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» 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.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).