- 6.4.1: Find x and y 35 x4 = 3y -34
- 6.4.2: Find x and y B 6x25R = B -95yR
- 6.4.3: Find x and y B 3y2x-8R = B 31-2-8R
- 6.4.4: Find x and y x - 1y + 34-7R = B 0-24-7
- 6.4.5: Find each of the following. A + B
- 6.4.6: Find each of the following. B + A
- 6.4.7: Find each of the following. E + 0
- 6.4.8: Find each of the following. 2A
- 6.4.9: Find each of the following.3F
- 6.4.10: Find each of the following.1-12D
- 6.4.11: Find each of the following.3F + 2A
- 6.4.12: Find each of the following.A - B
- 6.4.13: Find each of the following.B - A
- 6.4.14: Find each of the following.AB
- 6.4.15: Find each of the following.BA
- 6.4.16: Find each of the following.0F
- 6.4.17: Find each of the following.CD
- 6.4.18: Find each of the following.EF
- 6.4.19: Find each of the following.AI
- 6.4.20: Find each of the following.IA
- 6.4.21: Find the product, if possible. B -130-572R C6-41S
- 6.4.22: Find the product, if possible. 36 -1 24 C1-2540-3
- 6.4.23: Find the product, if possible. C-25-141-3S B 3-1-64
- 6.4.24: Find the product, if possible. B 20-1504R C-3051200-14S
- 6.4.25: Find the product, if possible. C1-53S B -60548-1
- 6.4.26: Find the product, if possible. C2000-10003S C02-1-410306
- 6.4.27: Find the product, if possible. 10-2-48-1305S C3000-40001
- 6.4.28: Find the product, if possible. B 4-5R C2600-7-3S
- 6.4.29: Produce. The produce manager at Dugans Marketorders 40 lb of tomato...
- 6.4.30: Budget. For the month of June, Madelyn budgets$400 for food, $160 f...
- 6.4.31: Nutrition. A 3-oz serving of roasted, skinlesschicken breast contai...
- 6.4.32: Nutrition. One slice of cheese pizza contains290 Cal, 15 g of prote...
- 6.4.33: Food Service Management. The food servicemanager at a large hospita...
- 6.4.34: Food Service Management. A college food servicemanager uses a table...
- 6.4.35: Profit. A manufacturer produces exterior plywood,interior plywood, ...
- 6.4.36: Production Cost. Karin supplies two smallcampus coffee shops with h...
- 6.4.37: Production Cost. In Exercise 35, suppose thatthe manufacturers prod...
- 6.4.38: Profit. In Exercise 36, suppose that Karins profitson one dozen cho...
- 6.4.39: Write a matrix equation equivalent to the system ofequations. 2x - ...
- 6.4.40: Write a matrix equation equivalent to the system ofequations. -x + ...
- 6.4.41: Write a matrix equation equivalent to the system ofequations. x + y...
- 6.4.42: Write a matrix equation equivalent to the system ofequations. 3x - ...
- 6.4.43: Write a matrix equation equivalent to the system ofequations. 3x - ...
- 6.4.44: Write a matrix equation equivalent to the system ofequations. 3x + ...
- 6.4.45: Write a matrix equation equivalent to the system ofequations. -4w +...
- 6.4.46: Write a matrix equation equivalent to the system ofequations. 12w +...
- 6.4.47: In Exercises 4750: a) Find the vertex.b) Find the axis of symmetry....
- 6.4.48: In Exercises 4750: a) Find the vertex.b) Find the axis of symmetry....
- 6.4.49: In Exercises 4750: a) Find the vertex.b) Find the axis of symmetry....
- 6.4.50: In Exercises 4750: a) Find the vertex.b) Find the axis of symmetry....
- 6.4.51: Show that1A + B21A - B2 A2 - B2,whereA2 = AA and B2 = BB.
- 6.4.52: Show that1A + B21A + B2 A2 + 2AB + B2.
- 6.4.53: Show that1A + B21A - B2 = A2 + BA - AB - B2
- 6.4.54: Show that1A + B21A + B2 = A2 + BA + AB + B2.
Solutions for Chapter 6.4: Matrix Operations
Full solutions for College Algebra: Graphs and Models | 5th Edition
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.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
cond(A) = c(A) = IIAIlIIA-III = 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.
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.
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.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Invert A by row operations on [A I] to reach [I A-I].
Gram-Schmidt 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.
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.
lA-II = 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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
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).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().