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# Solutions for Chapter 6.4: Multiplicative Inverses of Matrices and Matrix Equations

## Full solutions for College Algebra | 6th Edition

ISBN: 9780321782281

Solutions for Chapter 6.4: Multiplicative Inverses of Matrices and Matrix Equations

Solutions for Chapter 6.4
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##### ISBN: 9780321782281

This expansive textbook survival guide covers the following chapters and their solutions. Since 95 problems in chapter 6.4: Multiplicative Inverses of Matrices and Matrix Equations have been answered, more than 36989 students have viewed full step-by-step solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321782281. This textbook survival guide was created for the textbook: College Algebra , edition: 6. Chapter 6.4: Multiplicative Inverses of Matrices and Matrix Equations includes 95 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• Back substitution.

Upper triangular systems are solved in reverse order Xn to Xl.

• Block matrix.

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• Cofactor Cij.

Remove row i and column j; multiply the determinant by (-I)i + j •

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Hankel matrix H.

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

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Incidence matrix of a directed graph.

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

• Kirchhoff's Laws.

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

• 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.

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Pseudoinverse A+ (Moore-Penrose inverse).

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

• 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.

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

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