 2.3.1: In Exercises 14, show that is the inverse of A
 2.3.2: In Exercises 14, show that is the inverse of A
 2.3.3: In Exercises 14, show that is the inverse of A
 2.3.4: In Exercises 14, show that is the inverse of A
 2.3.5: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.6: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.7: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.8: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.9: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.10: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.11: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.12: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.13: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.14: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.15: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.16: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.17: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.18: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.19: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.20: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.21: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.22: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.23: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.24: In Exercises 524, find the inverse of the matrix (if it exists).
 2.3.25: In Exercises 2528, use an inverse matrix to solve each system of li...
 2.3.26: In Exercises 2528, use an inverse matrix to solve each system of li...
 2.3.27: In Exercises 2528, use an inverse matrix to solve each system of li...
 2.3.28: In Exercises 2528, use an inverse matrix to solve each system of li...
 2.3.29: In Exercises 2932, use a graphing utility or computer software prog...
 2.3.30: In Exercises 2932, use a graphing utility or computer software prog...
 2.3.31: In Exercises 2932, use a graphing utility or computer software prog...
 2.3.32: In Exercises 2932, use a graphing utility or computer software prog...
 2.3.33: In Exercises 3336, use the inverse matrices to find (a) (b) (c) and
 2.3.34: In Exercises 3336, use the inverse matrices to find (a) (b) (c) and
 2.3.35: In Exercises 3336, use the inverse matrices to find (a) (b) (c) and
 2.3.36: In Exercises 3336, use the inverse matrices to find (a) (b) (c) and
 2.3.37: In Exercises 37 and 38, find x such that the matrix is equal to its...
 2.3.38: In Exercises 37 and 38, find x such that the matrix is equal to its...
 2.3.39: In Exercises 39 and 40, find x such that the matrix is singular.
 2.3.40: In Exercises 39 and 40, find x such that the matrix is singular.
 2.3.41: In Exercises 41 and 42, find A provided that
 2.3.42: In Exercises 41 and 42, find A provided that
 2.3.43: In Exercises 43 and 44, show that the matrix is invertible and find...
 2.3.44: In Exercises 43 and 44, show that the matrix is invertible and find...
 2.3.45: (a) The inverse of a nonsingular matrix is unique. (b) If the matri...
 2.3.46: (a) The product of four invertible matrices is invertible. (b) The ...
 2.3.47: Prove Property 2 of Theorem 2.8: If is an invertible matrix and is ...
 2.3.48: Prove Property 4 of Theorem 2.8: If is an invertible matrix, then
 2.3.49: Guided Proof Prove that the inverse of a symmetric nonsingular matr...
 2.3.50: Prove Property 2 of Theorem 2.10: If is an invertible matrix such t...
 2.3.51: Prove that if then
 2.3.52: Prove that if and are square matrices and then is invertible and
 2.3.53: Prove that if is invertible and then
 2.3.54: Guided Proof Prove that if then either or is singular. Getting Star...
 2.3.55: Writing Is the sum of two invertible matrices invertible? Explain w...
 2.3.56: Writing Under what conditions will the diagonal matrix be invertibl...
 2.3.57: Use the result of Exercise 56 to find for each matrix. (a) (b)
 2.3.58: Let (a) Show that where is the identity matrix of order 2. (b) Show...
 2.3.59: Let be an column matrix satisfying The matrix is called a Household...
 2.3.60: Prove that if the matrix is nonsingular, then so is
 2.3.61: Let and be matrices satisfying If is nonsingular, solve this equati...
 2.3.62: Let and be matrices satisfying Solve this equation for Must it be t...
 2.3.63: Find an example of a singular 2 _ 2 matrix satisfying A2 _ A.
Solutions for Chapter 2.3: The Inverse of a Matrix
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 2.3: The Inverse of a Matrix
Get Full SolutionsSince 63 problems in chapter 2.3: The Inverse of a Matrix have been answered, more than 19540 students have viewed full stepbystep solutions from this chapter. Chapter 2.3: The Inverse of a Matrix includes 63 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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

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

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

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

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.