 8.7.1: In Exercises 1 and 2, fill in the blank. 1. Both det(A) and A repre...
 8.7.2: In Exercises 1 and 2, fill in the blank. 2. The determinant of the ...
 8.7.3: For a square matrix B, the minor M23 = 5. What is the cofactor C23 ...
 8.7.4: To find the determinant of a matrix using expanding by cofactors, d...
 8.7.5: In Exercises 512, find the determinant of the matrix. 5. [4]
 8.7.6: In Exercises 512, find the determinant of the matrix. 6. [12]
 8.7.7: In Exercises 512, find the determinant of the matrix. 7. [ 8 2 4 3]...
 8.7.8: In Exercises 512, find the determinant of the matrix. 8. [ 5 6 2 3]
 8.7.9: In Exercises 512, find the determinant of the matrix. 9. [ 7 1 2 6 3]
 8.7.10: In Exercises 512, find the determinant of the matrix. 10. [ 4 0 3 0...
 8.7.11: In Exercises 512, find the determinant of the matrix. 11. [ 3 4 3 3...
 8.7.12: In Exercises 512, find the determinant of the matrix. 12. [ 9 5 5 4]
 8.7.13: In Exercises 13 16, use the matrix capabilities of a graphing utili...
 8.7.14: In Exercises 13 16, use the matrix capabilities of a graphing utili...
 8.7.15: In Exercises 13 16, use the matrix capabilities of a graphing utili...
 8.7.16: In Exercises 13 16, use the matrix capabilities of a graphing utili...
 8.7.17: In Exercises 1720, find all (a) minors and (b) cofactors of the mat...
 8.7.18: In Exercises 1720, find all (a) minors and (b) cofactors of the mat...
 8.7.19: In Exercises 1720, find all (a) minors and (b) cofactors of the mat...
 8.7.20: In Exercises 1720, find all (a) minors and (b) cofactors of the mat...
 8.7.21: In Exercises 2124, find the determinant of the matrix. Expand by co...
 8.7.22: In Exercises 2124, find the determinant of the matrix. Expand by co...
 8.7.23: In Exercises 2124, find the determinant of the matrix. Expand by co...
 8.7.24: In Exercises 2124, find the determinant of the matrix. Expand by co...
 8.7.25: In Exercises 2534, find the determinant of the matrix. Expand by co...
 8.7.26: In Exercises 2534, find the determinant of the matrix. Expand by co...
 8.7.27: In Exercises 2534, find the determinant of the matrix. Expand by co...
 8.7.28: In Exercises 2534, find the determinant of the matrix. Expand by co...
 8.7.29: In Exercises 2534, find the determinant of the matrix. Expand by co...
 8.7.30: In Exercises 2534, find the determinant of the matrix. Expand by co...
 8.7.31: In Exercises 2534, find the determinant of the matrix. Expand by co...
 8.7.32: In Exercises 2534, find the determinant of the matrix. Expand by co...
 8.7.33: In Exercises 2534, find the determinant of the matrix. Expand by co...
 8.7.34: In Exercises 2534, find the determinant of the matrix. Expand by co...
 8.7.35: In Exercises 3538, use the matrix capabilities of a graphing utilit...
 8.7.36: In Exercises 3538, use the matrix capabilities of a graphing utilit...
 8.7.37: In Exercises 3538, use the matrix capabilities of a graphing utilit...
 8.7.38: In Exercises 3538, use the matrix capabilities of a graphing utilit...
 8.7.39: In Exercises 3942, find (a) A, (b) B, (c) AB, and (d) AB. What do y...
 8.7.40: In Exercises 3942, find (a) A, (b) B, (c) AB, and (d) AB. What do y...
 8.7.41: In Exercises 3942, find (a) A, (b) B, (c) AB, and (d) AB. What do y...
 8.7.42: In Exercises 3942, find (a) A, (b) B, (c) AB, and (d) AB. What do y...
 8.7.43: In Exercises 43 and 44, use the matrix capabilities of a graphing u...
 8.7.44: In Exercises 43 and 44, use the matrix capabilities of a graphing u...
 8.7.45: In Exercises 4550, evaluate the determinants to verify the equation...
 8.7.46: In Exercises 4550, evaluate the determinants to verify the equation...
 8.7.47: In Exercises 4550, evaluate the determinants to verify the equation...
 8.7.48: In Exercises 4550, evaluate the determinants to verify the equation...
 8.7.49: In Exercises 4550, evaluate the determinants to verify the equation...
 8.7.50: In Exercises 4550, evaluate the determinants to verify the equation...
 8.7.51: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.52: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.53: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.54: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.55: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.56: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.57: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.58: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.59: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.60: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.61: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.62: In Exercises 5162, solve for x. 51. x 1 2 x = 2 52. x 1 4 x = 20 53...
 8.7.63: In Exercises 6368, evaluate the determinant, in which the entries a...
 8.7.64: In Exercises 6368, evaluate the determinant, in which the entries a...
 8.7.65: In Exercises 6368, evaluate the determinant, in which the entries a...
 8.7.66: In Exercises 6368, evaluate the determinant, in which the entries a...
 8.7.67: In Exercises 6368, evaluate the determinant, in which the entries a...
 8.7.68: In Exercises 6368, evaluate the determinant, in which the entries a...
 8.7.69: In Exercises 69 and 70, determine whether the statement is true or ...
 8.7.70: In Exercises 69 and 70, determine whether the statement is true or ...
 8.7.71: Find a pair of 3 3 matrices A and B to demonstrate that A + B A + B.
 8.7.72: Let A be a 3 3 matrix such that A = 5. Can you use this information...
 8.7.73: In Exercises 7376, (a) find the determinant of A, (b) find A1, (c) ...
 8.7.74: In Exercises 7376, (a) find the determinant of A, (b) find A1, (c) ...
 8.7.75: In Exercises 7376, (a) find the determinant of A, (b) find A1, (c) ...
 8.7.76: In Exercises 7376, (a) find the determinant of A, (b) find A1, (c) ...
 8.7.77: In Exercises 7779, a property of determinants is given (A and B are...
 8.7.78: In Exercises 7779, a property of determinants is given (A and B are...
 8.7.79: In Exercises 7779, a property of determinants is given (A and B are...
 8.7.80: A diagonal matrix is a square matrix with all zero entries above an...
 8.7.81: A triangular matrix is a square matrix with all zero entries either...
 8.7.82: Explain why the determinant of the matrix is equal to zero. [ 2 1 0...
 8.7.83: Describe the different methods you have learned for finding the det...
 8.7.84: Consider square matrices in which the entries are consecutive integ...
 8.7.85: In Exercises 85 and 86, factor the expression. 85. 4y2 12y + 9 86. ...
 8.7.86: In Exercises 85 and 86, factor the expression. 85. 4y2 12y + 9 86. ...
 8.7.87: In Exercises 87 and 88, solve the system of equations using the met...
 8.7.88: In Exercises 87 and 88, solve the system of equations using the met...
Solutions for Chapter 8.7: Linear Systems and Matrices
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 8.7: Linear Systems and Matrices
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. Since 88 problems in chapter 8.7: Linear Systems and Matrices have been answered, more than 66056 students have viewed full stepbystep solutions from this chapter. Chapter 8.7: Linear Systems and Matrices includes 88 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

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).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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 .

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.

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

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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 ().

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.