 2.3.2.3.1: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.1: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.2.3.2: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.2: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.2.3.3: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.3: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.2.3.4: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.4: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.2.3.5: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.5: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.2.3.6: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.6: The Inverse of a Matrix In Exercises 16, show that B is the inverse...
 2.3.2.3.7: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.7: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.8: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.8: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.9: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.9: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.10: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.10: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.11: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.11: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.12: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.12: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.13: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.13: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.14: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.14: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.15: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.15: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.16: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.16: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.17: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.17: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.18: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.18: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.19: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.19: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.20: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.20: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.21: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.21: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.22: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.22: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.23: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.23: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.24: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.24: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.25: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.25: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.26: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.26: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.27: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.27: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.28: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.28: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.29: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.29: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.30: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.30: Finding the Inverse of a Matrix In Exercises 730, find the inverse ...
 2.3.2.3.31: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.31: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.2.3.32: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.32: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.2.3.33: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.33: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.2.3.34: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.34: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.2.3.35: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.35: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.2.3.36: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.36: Finding the Inverse of a 2 2 Matrix In Exercises 3136, use the form...
 2.3.2.3.37: Finding the Inverse of the Square of a Matrix In Exercises 3740, co...
 2.3.37: Finding the Inverse of the Square of a Matrix In Exercises 3740, co...
 2.3.2.3.38: Finding the Inverse of the Square of a Matrix In Exercises 3740, co...
 2.3.38: Finding the Inverse of the Square of a Matrix In Exercises 3740, co...
 2.3.2.3.39: Finding the Inverse of the Square of a Matrix In Exercises 3740, co...
 2.3.39: Finding the Inverse of the Square of a Matrix In Exercises 3740, co...
 2.3.2.3.40: Finding the Inverse of the Square of a Matrix In Exercises 3740, co...
 2.3.40: Finding the Inverse of the Square of a Matrix In Exercises 3740, co...
 2.3.2.3.41: Finding the Inverses of Products and Transposes In Exercises 4144, ...
 2.3.41: Finding the Inverses of Products and Transposes In Exercises 4144, ...
 2.3.2.3.42: Finding the Inverses of Products and Transposes In Exercises 4144, ...
 2.3.42: Finding the Inverses of Products and Transposes In Exercises 4144, ...
 2.3.2.3.43: Finding the Inverses of Products and Transposes In Exercises 4144, ...
 2.3.43: Finding the Inverses of Products and Transposes In Exercises 4144, ...
 2.3.2.3.44: Finding the Inverses of Products and Transposes In Exercises 4144, ...
 2.3.44: Finding the Inverses of Products and Transposes In Exercises 4144, ...
 2.3.2.3.45: Solving a System of Equations Using an Inverse In Exercises 4548, u...
 2.3.45: Solving a System of Equations Using an Inverse In Exercises 4548, u...
 2.3.2.3.46: Solving a System of Equations Using an Inverse In Exercises 4548, u...
 2.3.46: Solving a System of Equations Using an Inverse In Exercises 4548, u...
 2.3.2.3.47: Solving a System of Equations Using an Inverse In Exercises 4548, u...
 2.3.47: Solving a System of Equations Using an Inverse In Exercises 4548, u...
 2.3.2.3.48: Solving a System of Equations Using an Inverse In Exercises 4548, u...
 2.3.48: Solving a System of Equations Using an Inverse In Exercises 4548, u...
 2.3.2.3.49: Solving a System of Equations Using an Inverse In Exercises 4952, u...
 2.3.49: Solving a System of Equations Using an Inverse In Exercises 4952, u...
 2.3.2.3.50: Solving a System of Equations Using an Inverse In Exercises 4952, u...
 2.3.50: Solving a System of Equations Using an Inverse In Exercises 4952, u...
 2.3.2.3.51: Solving a System of Equations Using an Inverse In Exercises 4952, u...
 2.3.51: Solving a System of Equations Using an Inverse In Exercises 4952, u...
 2.3.2.3.52: Solving a System of Equations Using an Inverse In Exercises 4952, u...
 2.3.52: Solving a System of Equations Using an Inverse In Exercises 4952, u...
 2.3.2.3.53: Matrix Equal to Its Own Inverse In Exercises 53 and 54, find x such...
 2.3.53: Matrix Equal to Its Own Inverse In Exercises 53 and 54, find x such...
 2.3.2.3.54: Matrix Equal to Its Own Inverse In Exercises 53 and 54, find x such...
 2.3.54: Matrix Equal to Its Own Inverse In Exercises 53 and 54, find x such...
 2.3.2.3.55: Singular Matrix In Exercises 55 and 56, find x such that the matrix...
 2.3.55: Singular Matrix In Exercises 55 and 56, find x such that the matrix...
 2.3.2.3.56: Singular Matrix In Exercises 55 and 56, find x such that the matrix...
 2.3.56: Singular Matrix In Exercises 55 and 56, find x such that the matrix...
 2.3.2.3.57: Solving a Matrix Equation In Exercises 57 and 58, find A.(2A)1 = [1...
 2.3.57: Solving a Matrix Equation In Exercises 57 and 58, find A.(2A)1 = [1...
 2.3.2.3.58: Solving a Matrix Equation In Exercises 57 and 58, find A.(4A)1 = [ ...
 2.3.58: Solving a Matrix Equation In Exercises 57 and 58, find A.(4A)1 = [ ...
 2.3.2.3.59: Finding the Inverse of a Matrix In Exercises 59 and 60, show that t...
 2.3.59: Finding the Inverse of a Matrix In Exercises 59 and 60, show that t...
 2.3.2.3.60: Finding the Inverse of a Matrix In Exercises 59 and 60, show that t...
 2.3.60: Finding the Inverse of a Matrix In Exercises 59 and 60, show that t...
 2.3.2.3.61: Beam Deflection In Exercises 61 and 62, forces w1, w2, and w3 (in p...
 2.3.61: Beam Deflection In Exercises 61 and 62, forces w1, w2, and w3 (in p...
 2.3.2.3.62: Beam Deflection In Exercises 61 and 62, forces w1, w2, and w3 (in p...
 2.3.62: Beam Deflection In Exercises 61 and 62, forces w1, w2, and w3 (in p...
 2.3.2.3.63: Proof Prove Property 2 of Theorem 2.8: If A is aninvertible matrix ...
 2.3.63: Proof Prove Property 2 of Theorem 2.8: If A is aninvertible matrix ...
 2.3.2.3.64: Proof Prove Property 4 of Theorem 2.8: If A is aninvertible matrix,...
 2.3.64: Proof Prove Property 4 of Theorem 2.8: If A is aninvertible matrix,...
 2.3.2.3.65: Proof Prove Property 2 of Theorem 2.10: If C is aninvertible matrix...
 2.3.65: Proof Prove Property 2 of Theorem 2.10: If C is aninvertible matrix...
 2.3.2.3.66: Proof Prove that if A2 = A, thenI 2A = (I 2A)1.
 2.3.66: Proof Prove that if A2 = A, thenI 2A = (I 2A)1.
 2.3.2.3.67: Guided Proof Prove that the inverse of a symmetricnonsingular matri...
 2.3.67: Guided Proof Prove that the inverse of a symmetricnonsingular matri...
 2.3.2.3.68: Proof Prove that if A, B, and C are square matricesand ABC = I, the...
 2.3.68: Proof Prove that if A, B, and C are square matricesand ABC = I, the...
 2.3.2.3.69: Proof Prove that if A is invertible and AB = O,then B = O
 2.3.69: Proof Prove that if A is invertible and AB = O,then B = O
 2.3.2.3.70: Guided Proof Prove that if A2 = A, then either A issingular or A = ...
 2.3.70: Guided Proof Prove that if A2 = A, then either A issingular or A = ...
 2.3.2.3.71: True or False? In Exercises 71 and 72, determine whether each state...
 2.3.71: True or False? In Exercises 71 and 72, determine whether each state...
 2.3.2.3.72: True or False? In Exercises 71 and 72, determine whether each state...
 2.3.72: True or False? In Exercises 71 and 72, determine whether each state...
 2.3.2.3.73: Writing Is the sum of two invertible matrices invertible? Explain w...
 2.3.73: Writing Is the sum of two invertible matrices invertible? Explain w...
 2.3.2.3.74: Writing Under what conditions will the diagonal matrixA =[a11000a22...
 2.3.74: Writing Under what conditions will the diagonal matrixA =[a11000a22...
 2.3.2.3.75: Use the result of Exercise 74 to find A1 for eachmatrix.(a) A = [10...
 2.3.75: Use the result of Exercise 74 to find A1 for eachmatrix.(a) A = [10...
 2.3.2.3.76: Let A = [ 1221].(a) Show that A2 2A + 5I = O, where I is theidentit...
 2.3.76: Let A = [ 1221].(a) Show that A2 2A + 5I = O, where I is theidentit...
 2.3.2.3.77: Proof Let u be an n 1 column matrix satisfyinguTu = I1. The n n mat...
 2.3.77: Proof Let u be an n 1 column matrix satisfyinguTu = I1. The n n mat...
 2.3.2.3.78: Proof Let A and B be n n matrices. Prove that if thematrix I AB is ...
 2.3.78: Proof Let A and B be n n matrices. Prove that if thematrix I AB is ...
 2.3.2.3.79: Let A, D, and P be n n matrices satisfying AP = PD.Assume that P is...
 2.3.79: Let A, D, and P be n n matrices satisfying AP = PD.Assume that P is...
 2.3.2.3.80: Find an example of a singular 2 2 matrix satisfyingA2 = A.
 2.3.80: Find an example of a singular 2 2 matrix satisfyingA2 = A.
 2.3.2.3.81: Writing Explain how to determine whether the inverseof a matrix exi...
 2.3.81: Writing Explain how to determine whether the inverseof a matrix exi...
 2.3.2.3.82: CAPSTONE As mentioned on page 66, if Ais a 2 2 matrixA = [acbd] the...
 2.3.82: CAPSTONE As mentioned on page 66, if Ais a 2 2 matrixA = [acbd] the...
 2.3.2.3.83: Writing Explain in your own words how to write asystem of three lin...
 2.3.83: Writing Explain in your own words how to write asystem of three lin...
Solutions for Chapter 2.3: The Inverse of a Matrix
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 2.3: The Inverse of a Matrix
Get Full SolutionsSince 166 problems in chapter 2.3: The Inverse of a Matrix have been answered, more than 44270 students have viewed full stepbystep solutions from this chapter. Chapter 2.3: The Inverse of a Matrix includes 166 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8.

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

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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).

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

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.

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).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

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