Ch. 6 - Matrices and Determinants

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 27ab

Chapter 7, Problem 27ab

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 3 3 - 2 A = B = 5 3 - 1 6

Verified step by step guidance

Step 1: Identify the dimensions of matrices A and B. Matrix A is a 2x2 matrix (2 rows and 2 columns), and matrix B is also a 2x2 matrix.

Step 2: To find the product AB, multiply matrix A by matrix B. Since both are 2x2, the product AB is defined. Use the formula for matrix multiplication: the element in row i, column j of AB is the sum of the products of elements from row i of A and column j of B. Mathematically,

(AB)_ij = \sum_{k=1}^{n} A_ik B_kj

where n is the number of columns in A (or rows in B).

Step 3: To find the product BA, multiply matrix B by matrix A. Since both are 2x2 matrices, the product BA is also defined. Use the same matrix multiplication rule as in Step 2, but with B as the first matrix and A as the second.

Step 4: Perform the multiplication for each element of AB and BA by calculating the sum of products for each corresponding row and column.

Step 5: Write the resulting matrices AB and BA after completing the calculations for each element.

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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 number of columns in the first matrix must equal the number of rows in the second matrix for the product to be defined.

Matrix Dimensions and Compatibility

To multiply two matrices A and B, the number of columns in A must match the number of rows in B. The resulting matrix has dimensions equal to the number of rows of A and the number of columns of B.

Non-Commutativity of Matrix Multiplication

Matrix multiplication is generally not commutative, meaning AB does not necessarily equal BA. Both products may exist or only one may be defined, depending on the dimensions of the matrices.

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Textbook Question

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\begin{vmatrix}\)1 & 1 & 1 \\2 & 2 & 2 \\-3 & 4 & -5\(\end{vmatrix}$

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Textbook Question

Solve for X in the matrix equation 3X+A = B where

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Textbook Question

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\begin{vmatrix}\)0.5 & 7 & 5 \\0.5 & 3 & 9 \\0.5 & 1 & 3\(\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.

$\begin{cases}\)x + y + z = 4 $\x$ - y - z = 0 $\x$ - y + z = 2\(\end{cases}$

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Textbook Question

Let $A = [\begin{matrix} - 3 & - 7 \\ 2 & - 9 \\ 5 & 0 \end{matrix}]$ and $B = [\begin{matrix} - 5 & - 1 \\ 0 & 0 \\ 3 & - 4 \end{matrix}]$ . Solve each matrix equation for X. 4A + 3B = - 2X

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Textbook Question

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

$\begin{cases}\)x + 2y = z - 1 $\x$ = 4 + y - z $\x$ + y - 3z = -2\(\end{cases}$

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