 4.2.1: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.2: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.3: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.4: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.5: Write two matrices that represent these data for males and females.
 4.2.6: Find the total number of students that participate in each individu...
 4.2.7: Could you add the two matrices to find the total number of schools ...
 4.2.8: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.9: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.10: Use matrices A, B, C, and D to find the following.
 4.2.11: Use matrices A, B, C, and D to find the following.
 4.2.12: Use matrices A, B, C, and D to find the following.
 4.2.13: Use matrices A, B, C, and D to find the following.
 4.2.14: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.15: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.16: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.17: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.18: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.19: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.20: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.21: Perform the indicated matrix operations. If the matrix does not exi...
 4.2.22: For Exercises 2224, use the following information. An electronics s...
 4.2.23: For Exercises 2224, use the following information. An electronics s...
 4.2.24: For Exercises 2224, use the following information. An electronics s...
 4.2.25: Perform the indicated matrix operation. If the matrix does not exis...
 4.2.26: Perform the indicated matrix operation. If the matrix does not exis...
 4.2.27: Perform the indicated matrix operation. If the matrix does not exis...
 4.2.28: Perform the indicated matrix operation. If the matrix does not exis...
 4.2.29: Use matrices A, B, C, and D to find the following
 4.2.30: Use matrices A, B, C, and D to find the following
 4.2.31: Use matrices A, B, C, and D to find the following
 4.2.32: Use matrices A, B, C, and D to find the following
 4.2.33: Use matrices A, B, C, and D to find the following
 4.2.34: Use matrices A, B, C, and D to find the following
 4.2.35: Perform the indicated matrix operation. If the matrix does not exis...
 4.2.36: Perform the indicated matrix operation. If the matrix does not exis...
 4.2.37: Perform the indicated matrix operation. If the matrix does not exis...
 4.2.38: Perform the indicated matrix operation. If the matrix does not exis...
 4.2.39: Find the difference between U.S. and World records expressed as a c...
 4.2.40: Write a matrix that compares the total time of all four events for ...
 4.2.41: In which events were the fastest times set at the Olympics?
 4.2.42: Write the matrix that represents the additional cost for nonresidents
 4.2.43: Write a matrix that represents the difference in cost if a child or...
 4.2.44: Determine values for each variable if d = 1, e = 4d, z + d = e, f =...
 4.2.45: Give an example of two matrices whose sum is a zero matrix
 4.2.46: For matrix A = 1 3 2 4 , the transpose of A is AT = 1 2 3 4 . Write...
 4.2.47: Use the data on nutrition on page 169 to explain how matrices can b...
 4.2.48: Solve for x and y in the matrix equation x 7 + 3y x = 16 12 . A x ...
 4.2.49: What is the equation of the line that has a slope of 3 and passes t...
 4.2.50: State the dimensions of each matrix.
 4.2.51: State the dimensions of each matrix.
 4.2.52: State the dimensions of each matrix.
 4.2.53: State the dimensions of each matrix.
 4.2.54: State the dimensions of each matrix.
 4.2.55: State the dimensions of each matrix.
 4.2.56: Solve each system of equations
 4.2.57: Solve each system of equations
 4.2.58: Solve each system of equations
 4.2.59: Solve each system by using substitution or elimination.
 4.2.60: Solve each system by using substitution or elimination.
 4.2.61: Solve each system by using substitution or elimination.
 4.2.62: Write and graph an inequality that describes this situation.
 4.2.63: Does Ian have enough money to buy 14 pieces of each type of paper? ...
 4.2.64: Name the property illustrated by each equation.
 4.2.65: Name the property illustrated by each equation.
 4.2.66: Name the property illustrated by each equation.
 4.2.67: Name the property illustrated by each equation.
Solutions for Chapter 4.2: Operations with Matrices
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 4.2: Operations with Matrices
Get Full SolutionsChapter 4.2: Operations with Matrices includes 67 full stepbystep solutions. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. Since 67 problems in chapter 4.2: Operations with Matrices have been answered, more than 44860 students have viewed full stepbystep solutions from this chapter.

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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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).

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!).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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).

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

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.