 1.7.1: In Exercises 12, classify the matrix as upper triangular, lower tri...
 1.7.2: In Exercises 12, classify the matrix as upper triangular, lower tri...
 1.7.3: In Exercises 36, find the product by inspection.300 0 1 0 002 2 1 4...
 1.7.4: In Exercises 36, find the product by inspection.1 2 5 3 1 0 400 030...
 1.7.5: In Exercises 36, find the product by inspection.500 020 0 0 3 3204 ...
 1.7.6: In Exercises 36, find the product by inspection. 200 0 1 0 004 4 1 ...
 1.7.7: In Exercises 710, find A2, A2, and Ak (where k is any integer) by i...
 1.7.8: In Exercises 710, find A2, A2, and Ak (where k is any integer) by i...
 1.7.9: In Exercises 710, find A2, A2, and Ak (where k is any integer) by i...
 1.7.10: In Exercises 710, find A2, A2, and Ak (where k is any integer) by i...
 1.7.11: In Exercises 1112, compute the product by inspection. 100 000 003 2...
 1.7.12: In Exercises 1112, compute the product by inspection. 100 020 004 3...
 1.7.13: In Exercises 1314, compute the indicated quantity.1 0 0 1 39
 1.7.14: In Exercises 1314, compute the indicated quantity.1 0 0 1 1000 I
 1.7.15: In Exercises 1516, use what you have learned in this section about ...
 1.7.16: In Exercises 1516, use what you have learned in this section about ...
 1.7.17: In Exercises 1718, create a symmetric matrix by substituting approp...
 1.7.18: In Exercises 1718, create a symmetric matrix by substituting approp...
 1.7.19: In Exercises 1922, determine by inspection whether the matrix is in...
 1.7.20: In Exercises 1922, determine by inspection whether the matrix is in...
 1.7.21: In Exercises 1922, determine by inspection whether the matrix is in...
 1.7.22: In Exercises 1922, determine by inspection whether the matrix is in...
 1.7.23: In Exercises 2324, find the diagonal entries of AB by inspection.A ...
 1.7.24: In Exercises 2324, find the diagonal entries of AB by inspection.A ...
 1.7.25: In Exercises 2526, find all values of the unknown constant(s) for w...
 1.7.26: In Exercises 2526, find all values of the unknown constant(s) for w...
 1.7.27: In Exercises 2728, find all values of x for which A is invertible.A...
 1.7.28: In Exercises 2728, find all values of x for which A is invertible. ...
 1.7.29: If A is an invertible upper triangular or lower triangular matrix, ...
 1.7.30: Show that if A is a symmetric n n matrix andB is any n m matrix, th...
 1.7.31: In Exercises 3132, find a diagonal matrix A that satisfies the give...
 1.7.32: In Exercises 3132, find a diagonal matrix A that satisfies the give...
 1.7.33: Verify Theorem 1.7.1(b) for the matrix product AB and Theorem 1.7.1...
 1.7.34: Let A be an n n symmetric matrix. (a) Show that A2 is symmetric. (b...
 1.7.35: Verify Theorem 1.7.4 for the given matrix A. (a) A = 2 1 1 3 (b) A ...
 1.7.36: Find all 3 3 diagonal matrices A that satisfy A2 3A 4I = 0.
 1.7.37: Let A = [aij ] be an n n matrix. Determine whether A is symmetric. ...
 1.7.38: On the basis of your experience with Exercise 37, devise a general ...
 1.7.39: Find an upper triangular matrix that satisfies A3 = 1 30 0 8
 1.7.40: If the n n matrix A can be expressed as A = LU, where L is a lower ...
 1.7.41: In the text we defined a matrix A to be symmetric if AT = A. Analog...
 1.7.42: In the text we defined a matrix A to be symmetric if AT = A. Analog...
 1.7.43: In the text we defined a matrix A to be symmetric if AT = A. Analog...
 1.7.44: In the text we defined a matrix A to be symmetric if AT = A. Analog...
 1.7.45: In the text we defined a matrix A to be symmetric if AT = A. Analog...
 1.7.46: Prove: If the matrices A and B are both upper triangular or both lo...
 1.7.47: Prove: If AT A = A, then A is symmetric and A = A2.
Solutions for Chapter 1.7: Diagonal,Triangular, and Symmetric Matrices
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 1.7: Diagonal,Triangular, and Symmetric Matrices
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. Chapter 1.7: Diagonal,Triangular, and Symmetric Matrices includes 47 full stepbystep solutions. Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228. Since 47 problems in chapter 1.7: Diagonal,Triangular, and Symmetric Matrices have been answered, more than 14941 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

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

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Special solutions to As = O.
One free variable is Si = 1, other free variables = 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.