 1.4.1: Let Show that (a) (b) (c) (d)
 1.4.2: Using the matrices and scalars in Exercise 1, verify that (a) (b) (...
 1.4.3: Using the matrices and scalars in Exercise 1, verify that (a) (b) (...
 1.4.4: In Exercises 47 use Theorem 1.4.5 to compute the inverses of the fo...
 1.4.5: In Exercises 47 use Theorem 1.4.5 to compute the inverses of the fo...
 1.4.6: In Exercises 47 use Theorem 1.4.5 to compute the inverses of the fo...
 1.4.7: In Exercises 47 use Theorem 1.4.5 to compute the inverses of the fo...
 1.4.8: Find the inverse of
 1.4.9: Find the inverse of
 1.4.10: Use the matrix A in Exercise 4 to verify that
 1.4.11: Use the matrix B in Exercise 5 to verify that
 1.4.12: Use the matrices A and B in 4 and 5 to verify that
 1.4.13: Use the matrices A, B, and C in Exercises 46 to verify that
 1.4.14: In Exercises 1417, use the given information to find A.
 1.4.15: In Exercises 1417, use the given information to find A.
 1.4.16: In Exercises 1417, use the given information to find A.
 1.4.17: In Exercises 1417, use the given information to find A.
 1.4.18: Let A be the matrix In each part, compute the given quantity. (a) (...
 1.4.19: Repeat Exercise 18 for the matrix
 1.4.20: Repeat Exercise 18 for the matrix
 1.4.21: Repeat Exercise 18 for the matrix
 1.4.22: In Exercises 2224, let , and . Show that for the given matrix. The ...
 1.4.23: In Exercises 2224, let , and . Show that for the given matrix.The m...
 1.4.24: In Exercises 2224, let , and . Show that for the given matrix.An ar...
 1.4.25: Show that if and then .
 1.4.26: Show that if and then .
 1.4.27: Consider the matrix where . Show that A is invertible and find its ...
 1.4.28: Show that if a square matrix A satisfies , then .
 1.4.29: (a) Show that a matrix with a row of zeros cannot have an inverse. ...
 1.4.30: Assuming that all matrices are and invertible, solve for
 1.4.31: Assuming that all matrices are and invertible, solve for D.
 1.4.32: If A is a square matrix and n is a positive integer, is it true tha...
 1.4.33: Simplify:
 1.4.34: Simplify:
 1.4.35: In Exercises 3537, determine whether A is invertible, and if so, fi...
 1.4.36: In Exercises 3537, determine whether A is invertible, and if so, fi...
 1.4.37: In Exercises 3537, determine whether A is invertible, and if so, fi...
 1.4.38: Prove Theorem 1.4.2.
 1.4.39: In Exercises 3942, use the method of Example 8 to find the unique s...
 1.4.40: In Exercises 3942, use the method of Example 8 to find the unique s...
 1.4.41: In Exercises 3942, use the method of Example 8 to find the unique s...
 1.4.42: In Exercises 3942, use the method of Example 8 to find the unique s...
 1.4.43: Prove part (a) of Theorem 1.4.1.
 1.4.44: Prove part (c) of Theorem 1.4.1.
 1.4.45: Prove part (f) of Theorem 1.4.1.
 1.4.46: Prove part (b) of Theorem 1.4.2.
 1.4.47: Prove part (c) of Theorem 1.4.2.
 1.4.48: Verify Formula 4 in the text by a direct calculation.
 1.4.49: Prove part (d) of Theorem 1.4.8.
 1.4.50: Prove part (e) of Theorem 1.4.8.
 1.4.51: (a) Show that if A is invertible and , then . (b) Explain why part ...
 1.4.52: Show that if A is invertible and k is any nonzero scalar, then for ...
 1.4.53: (a) Show that if A, B, and are invertible matrices with the same si...
 1.4.54: A square matrix A is said to be idempotent if . (a) Show that if A ...
 1.4.55: Show that if A is a square matrix such that for some positive integ...
 1.4.a: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
 1.4.b: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
 1.4.c: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
 1.4.d: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
 1.4.e: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
 1.4.f: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
 1.4.g: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
 1.4.h: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
 1.4.i: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
 1.4.j: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
 1.4.k: TrueFalse Exercises In parts (a)(k) determine whether the statemen...
Solutions for Chapter 1.4: Inverses; Algebraic Properties of Matrices
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 1.4: Inverses; Algebraic Properties of Matrices
Get Full SolutionsChapter 1.4: Inverses; Algebraic Properties of Matrices includes 66 full stepbystep solutions. Since 66 problems in chapter 1.4: Inverses; Algebraic Properties of Matrices have been answered, more than 14232 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. 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.

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.)

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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.

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