 8.6.1: In 1 and 2, verify that the matrix B is the inverse of the matrix A.
 8.6.2: In 1 and 2, verify that the matrix B is the inverse of the matrix A.
 8.6.3: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.4: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.5: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.6: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.7: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.8: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.9: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.10: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.11: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.12: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.13: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.14: In 314, use Theorem 8.6.3 to determine whether the given matrix is ...
 8.6.15: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.16: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.17: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.18: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.19: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.20: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.21: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.22: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.23: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.24: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.25: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.26: In 15  26, use Theorem 8.6.4 to find the inverse of the given matr...
 8.6.27: In 27 and 28, use the given matrices to find (AB) 1 .
 8.6.28: In 27 and 28, use the given matrices to find (AB) 1 .
 8.6.29: If A1 a 4 3 3 2b , what is A?
 8.6.30: If A is nonsingular, then (AT ) 1 (A1 ) T . Verify this for A a 1 4...
 8.6.31: Find a value of x such that the matrix A a 4 3 x 4 b is its own inv...
 8.6.32: Find the inverse of A a sinu cosu cosu sinu b .
 8.6.33: A nonsingular matrix A is said to be orthogonal if A1 AT . (a) Veri...
 8.6.34: Show that if A is an orthogonal matrix (see 33), then det A 1.
 8.6.35: Suppose A and B are nonsingular n 3 n matrices. Then show that AB i...
 8.6.36: Suppose A and B are n 3 n matrices and that either A or B is singul...
 8.6.37: Suppose A is a nonsingular matrix. Then show that det A1 1>det A.
 8.6.38: Suppose A2 A. Then show that either A I or A is singular
 8.6.39: Suppose A and B are n 3 n matrices, A is nonsingular, and AB 0. The...
 8.6.40: Suppose A and B are n 3 n matrices, A is nonsingular, and AB AC. Th...
 8.6.41: Suppose A and B are nonsingular n 3 n matrices. Is A B necessarily ...
 8.6.42: Suppose A is a nonsingular matrix. Then show that AT is nonsingular.
 8.6.43: Suppose A and B are n 3 n nonzero matrices and AB 0. Then show that...
 8.6.44: Consider the 3 3 diagonal matrix A a11 0 0 0 a22 0 0 0 a33 . Determ...
 8.6.45: In 45  52, use an inverse matrix to solve the given system of equa...
 8.6.46: In 45  52, use an inverse matrix to solve the given system of equa...
 8.6.47: In 45  52, use an inverse matrix to solve the given system of equa...
 8.6.48: In 45  52, use an inverse matrix to solve the given system of equa...
 8.6.49: In 45  52, use an inverse matrix to solve the given system of equa...
 8.6.50: In 45  52, use an inverse matrix to solve the given system of equa...
 8.6.51: In 45  52, use an inverse matrix to solve the given system of equa...
 8.6.52: In 45  52, use an inverse matrix to solve the given system of equa...
 8.6.53: In 53 and 54, write the system in the form AX B. Use X A1 B to solv...
 8.6.54: In 53 and 54, write the system in the form AX B. Use X A1 B to solv...
 8.6.55: In 5558, without solving, determine whether the given homogeneous s...
 8.6.56: In 5558, without solving, determine whether the given homogeneous s...
 8.6.57: In 5558, without solving, determine whether the given homogeneous s...
 8.6.58: In 5558, without solving, determine whether the given homogeneous s...
 8.6.59: The system of equations for the currents i1, i2, and i3 in the netw...
 8.6.60: Consider the square plate shown in FIGURE 8.6.2, with the temperatu...
Solutions for Chapter 8.6: Inverse of a Matrix
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 8.6: Inverse of a Matrix
Get Full SolutionsSince 60 problems in chapter 8.6: Inverse of a Matrix have been answered, more than 10742 students have viewed full stepbystep solutions from this chapter. Chapter 8.6: Inverse of a Matrix includes 60 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Advanced Engineering Mathematics was written by Patricia and is associated to the ISBN: 9781284105902.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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).

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

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).

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).
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