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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblem 31ab
Chapter 7, Problem 31ab

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 - 1 4 1 1 0 A = 4 - 1 3 B = 1 2 4 2 0 - 2 1 - 1 3
Matrices A and B for matrix operations in college algebra, chapter on matrices and determinants.

Verified step by step guidance
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Step 1: Identify the dimensions of matrices A and B. Matrix A is a 3x3 matrix (3 rows and 3 columns), and matrix B is also a 3x3 matrix.
Step 2: To find the product AB, multiply matrix A by matrix B. Since both are 3x3, the product AB is defined and will result in a 3x3 matrix. Each element (i,j) of AB is found by taking the dot product of the i-th row of A with the j-th column of B.
Step 3: To find the product BA, multiply matrix B by matrix A. Since both are 3x3, the product BA is also defined and will result in a 3x3 matrix. Each element (i,j) of BA is found by taking the dot product of the i-th row of B with the j-th column of A.
Step 4: For each element in the resulting matrices, perform the multiplication and addition as per the dot product rule: sum of products of corresponding elements from the row of the first matrix and the column of the second matrix.
Step 5: Write out the resulting matrices AB and BA by calculating each element step-by-step, but do not compute the final numerical values here.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. The element in the ith row and jth column of the product matrix is the dot product of the ith row of the first matrix and the jth column of the second matrix.
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Dimension Compatibility for Multiplication

Two matrices can be multiplied only if the number of columns in the first matrix equals the number of rows in the second matrix. For example, if A is m×n and B is p×q, multiplication AB is defined only if n = p, resulting in an m×q matrix.
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Non-Commutativity of Matrix Multiplication

Matrix multiplication is generally not commutative, meaning AB does not necessarily equal BA. Even if both products are defined, their results can differ, so both AB and BA must be computed separately to understand their values.
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Related Practice
Textbook Question

Write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.

{x+3y+4z=−3x+2y+3z=−2x+4y+3z=−6\(\begin{cases}\)x + 3y + 4z = -3 \(\x\) + 2y + 3z = -2 \(\x\) + 4y + 3z = -6\(\end{cases}\)

Textbook Question

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. ∣1561451910∣\(\begin{vmatrix}\)1 & 5 & 6 \\1 & 4 & 5 \\1 & 9 & 10\(\end{vmatrix}\)

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Textbook Question

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

{3a−b−4c=32a−b+2c=−8a+2b−3c=9\(\begin{cases}\)3a - b - 4c = 3 \\2a - b + 2c = -8 \(\a\) + 2b - 3c = 9\(\end{cases}\)

Textbook Question

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

{2x+2y+7z=−12x+y+2z=24x+6y+z=15\(\begin{cases}\)2x + 2y + 7z = -1 \\2x + y + 2z = 2 \\4x + 6y + z = 15\(\end{cases}\)

Textbook Question

Write each matrix equation as a system of linear equations without matrices.

[4−72−3][xy]=[−31]\(\begin{bmatrix}\)4 & -7 \\2 & -3\(\end{bmatrix}\[\begin{bmatrix}\)x \(\y\]\end{bmatrix}\)=\(\begin{bmatrix}\)-3 \\1\(\end{bmatrix}\)

Textbook Question

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. ∣−34−55−208−13∣\(\begin{vmatrix}\)-3 & 4 & -5 \\5 & -2 & 0 \\8 & -1 & 3\(\end{vmatrix}\)