 6.5.1: Determine whether B is the inverse of A. A = B 1237R , B = B 7231
 6.5.2: Determine whether B is the inverse of A. A = B 3423R , B = B 3423
 6.5.3: Determine whether B is the inverse of A. A = C111100623S , B = ...
 6.5.4: Determine whether B is the inverse of A. A = C253010374S , B = C...
 6.5.5: Use the GaussJordan method to find A1, if it exists.Check your ans...
 6.5.6: Use the GaussJordan method to find A1, if it exists.Check your ans...
 6.5.7: Use the GaussJordan method to find A1, if it exists.Check your ans...
 6.5.8: Use the GaussJordan method to find A1, if it exists.Check your ans...
 6.5.9: Use the GaussJordan method to find A1, if it exists.Check your ans...
 6.5.10: Use the GaussJordan method to find A1, if it exists.Check your ans...
 6.5.11: Use the GaussJordan method to find A1, if it exists.Check your ans...
 6.5.12: Use the GaussJordan method to find A1, if it exists.Check your ans...
 6.5.13: Use a graphing calculator to find A1, if it exists. A = B 4132R
 6.5.14: Use a graphing calculator to find A1, if it exists. A = B 0110
 6.5.15: Use a graphing calculator to find A1, if it exists. A = C231331241
 6.5.16: Use a graphing calculator to find A1, if it exists. A = C121213323
 6.5.17: Use a graphing calculator to find A1, if it exists. A = C121201110
 6.5.18: Use a graphing calculator to find A1, if it exists. A = C724100...
 6.5.19: Use a graphing calculator to find A1, if it exists. A = C724100...
 6.5.20: Use a graphing calculator to find A1, if it exists. A = C110011...
 6.5.21: Use a graphing calculator to find A1, if it exists. A = D100021003...
 6.5.22: Use a graphing calculator to find A1, if it exists. A = D200231...
 6.5.23: Use a graphing calculator to find A1, if it exists. A = D1111142...
 6.5.24: Use a graphing calculator to find A1, if it exists. A = D1037420...
 6.5.25: In Exercises 2528, a system of equations is given, togetherwith the...
 6.5.26: In Exercises 2528, a system of equations is given, togetherwith the...
 6.5.27: In Exercises 2528, a system of equations is given, togetherwith the...
 6.5.28: In Exercises 2528, a system of equations is given, togetherwith the...
 6.5.29: Solve the system of equations using the inverse of thecoefficient m...
 6.5.30: Solve the system of equations using the inverse of thecoefficient m...
 6.5.31: Solve the system of equations using the inverse of thecoefficient m...
 6.5.32: Solve the system of equations using the inverse of thecoefficient m...
 6.5.33: Solve the system of equations using the inverse of thecoefficient m...
 6.5.34: Solve the system of equations using the inverse of thecoefficient m...
 6.5.35: Solve the system of equations using the inverse of thecoefficient m...
 6.5.36: Solve the system of equations using the inverse of thecoefficient m...
 6.5.37: Solve the system of equations using the inverse of thecoefficient m...
 6.5.38: Solve the system of equations using the inverse of thecoefficient m...
 6.5.39: Sales. Kayla sold a total of 145 Italian sausagesand hot dogs from ...
 6.5.40: Price of School Supplies. Rubio bought 4 labrecord books and 3 high...
 6.5.41: Cost of Materials. GreenUp Landscapingbought 4 tons of topsoil, 3 ...
 6.5.42: Investment. Donna receives $230 per year in simpleinterest from thr...
 6.5.43: Use synthetic division to find the function values f 1x2 = x3  6x2...
 6.5.44: Use synthetic division to find the function values f 1x2 = 2x4  x3...
 6.5.45: Solve. 2x2 + x = 7
 6.5.46: Solve. 1x + 16x  1= 1
 6.5.47: Solve. 22x + 1  1 = 22x  4
 6.5.48: Solve. x  2x  6 = 0
 6.5.49: Factor the polynomial f 1x2. f 1x2 = x3  3x2  6x + 8
 6.5.50: Factor the polynomial f 1x2. f 1x2 = x4 + 2x3  16x2  2x + 15
 6.5.51: State the conditions under which A1 exists. Then find aformula for...
 6.5.52: State the conditions under which A1 exists. Then find aformula for...
 6.5.53: State the conditions under which A1 exists. Then find aformula for...
 6.5.54: State the conditions under which A1 exists. Then find aformula for...
Solutions for Chapter 6.5: Inverses of Matrices
Full solutions for College Algebra: Graphs and Models  5th Edition
ISBN: 9780321783950
Solutions for Chapter 6.5: Inverses of Matrices
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra: Graphs and Models, edition: 5. College Algebra: Graphs and Models was written by and is associated to the ISBN: 9780321783950. Chapter 6.5: Inverses of Matrices includes 54 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 54 problems in chapter 6.5: Inverses of Matrices have been answered, more than 28672 students have viewed full stepbystep solutions from this chapter.

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.)

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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).

Column space C (A) =
space of all combinations of the columns of A.

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 II· Condition numbers measure the sensitivity of the output to change in the input.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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 •

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

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!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.