 1.7.1: In Exercises 14, determine whether the given matrix is invertible.
 1.7.2: In Exercises 14, determine whether the given matrix is invertible.
 1.7.3: In Exercises 14, determine whether the given matrix is invertible.
 1.7.4: In Exercises 14, determine whether the given matrix is invertible.
 1.7.5: In Exercises 58, determine the product by inspection.
 1.7.6: In Exercises 58, determine the product by inspection.
 1.7.7: In Exercises 58, determine the product by inspection.
 1.7.8: In Exercises 58, determine the product by inspection.
 1.7.9: In Exercises 912, find , , and (where k is any integer) by inspection.
 1.7.10: In Exercises 912, find , , and (where k is any integer) by inspection.
 1.7.11: In Exercises 912, find , , and (where k is any integer) by inspection.
 1.7.12: In Exercises 912, find , , and (where k is any integer) by inspection.
 1.7.13: In Exercises 1319, decide whether the given matrix is symmetric.
 1.7.14: In Exercises 1319, decide whether the given matrix is symmetric.
 1.7.15: In Exercises 1319, decide whether the given matrix is symmetric.
 1.7.16: In Exercises 1319, decide whether the given matrix is symmetric.
 1.7.17: In Exercises 1319, decide whether the given matrix is symmetric.
 1.7.18: In Exercises 1319, decide whether the given matrix is symmetric.
 1.7.19: In Exercises 1319, decide whether the given matrix is symmetric.
 1.7.20: In Exercises 2022, decide by inspection whether the given matrix is...
 1.7.21: In Exercises 2022, decide by inspection whether the given matrix is...
 1.7.22: In Exercises 2022, decide by inspection whether the given matrix is...
 1.7.23: In Exercises 2324, find all values of the unknown constant(s) in or...
 1.7.24: In Exercises 2324, find all values of the unknown constant(s) in or...
 1.7.25: In Exercises 2526, find all values of x in order for A to be invert...
 1.7.26: In Exercises 2526, find all values of x in order for A to be invert...
 1.7.27: In Exercises 2728, find a diagonal matrix A that satisfies the give...
 1.7.28: In Exercises 2728, find a diagonal matrix A that satisfies the give...
 1.7.29: Verify Theorem 1.7.1(b) for the product AB, where
 1.7.30: Verify Theorem 1.7.1(d) for the matrices A and B in Exercise 29.
 1.7.31: Verify Theorem 1.7.4 for the given matrix A. (a) (b)
 1.7.32: Let A be an symmetric matrix. (a) Show that A2 is symmetric. (b) Sh...
 1.7.33: Prove: If , then A is symmetric and .
 1.7.34: Find all diagonal matrices A that satisfy .
 1.7.35: Let be an matrix. Determine whether A is symmetric. (a) (b) (c) (d)
 1.7.36: On the basis of your experience with Exercise 35, devise a general ...
 1.7.37: A square matrix A is called skewsymmetric if . Prove: (a) If A is ...
 1.7.38: In Exercises 3839, fill in the missing entries (marked with ) to pr...
 1.7.39: In Exercises 3839, fill in the missing entries (marked with ) to pr...
 1.7.40: Find all values of a, b, c, and d for which A is skewsymmetric.
 1.7.41: We showed in the text that the product of symmetric matrices is sym...
 1.7.42: If the matrix A can be expressed as , where L is a lower triangular...
 1.7.43: Find an upper triangular matrix that satisfies
 1.7.a: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.b: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.c: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.d: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.e: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.f: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.g: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.h: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.i: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.j: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.k: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.l: In parts (a)(m) determine whether the statement is true or false, a...
 1.7.m: In parts (a)(m) determine whether the statement is true or false, a...
Solutions for Chapter 1.7: Diagonal, Triangular, and Symmetric Matrices
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 1.7: Diagonal, Triangular, and Symmetric Matrices
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. Since 56 problems in chapter 1.7: Diagonal, Triangular, and Symmetric Matrices have been answered, more than 14056 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.7: Diagonal, Triangular, and Symmetric Matrices includes 56 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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

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.

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

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