Ch. 6 - Matrices and Determinants

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

Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 9b

Chapter 7, Problem 9b

In Exercises 9 - 16, find the following matrices: b. A - B

Verified step by step guidance

Step 1: Understand the problem. You are asked to find the matrix A - B, where matrices A and B are given as:

A= [\begin{matrix} 4 & 1 \\ 3 & 2 \end{matrix}]

and

B= [\begin{matrix} 5 & 9 \\ 0 & 7 \end{matrix}]

Step 2: Recall the rule for matrix subtraction. To subtract two matrices, subtract their corresponding elements. That is, if

A = [a_{ij}]

and

B = [b_{ij}]

, then

A - B = [a_{ij} - b_{ij}]

Step 3: Set up the subtraction element-wise. For each element in the resulting matrix, subtract the element in B from the corresponding element in A:

A - B = \(\begin{bmatrix}\) 4 - 5 & 1 - 9 \\ 3 - 0 & 2 - 7 \(\end{bmatrix}\)

Step 4: Write the resulting matrix with the subtracted elements (do not calculate the final values yet). This matrix represents the difference between A and B.

Step 5: Verify that both matrices A and B have the same dimensions (2x2), which is necessary for subtraction to be valid.

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Key Concepts

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

Matrix Subtraction

Matrix subtraction involves subtracting corresponding elements of two matrices of the same dimensions. Each element in the resulting matrix is found by subtracting the element in matrix B from the element in matrix A at the same position.

Matrix Dimensions and Compatibility

For matrix operations like addition or subtraction, the matrices must have the same dimensions (same number of rows and columns). Here, both matrices A and B are 2x2, making subtraction possible.

Representation of Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. Understanding how to read and write matrices is essential for performing operations like addition, subtraction, and multiplication.

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Related Practice

Textbook Question

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

Textbook Question

In Exercises 9 - 16, find the following matrices: c. - 4A

Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases}\)w - 2x - y - 3z = -9 $\w$ + x - y = 0 \\3w + 4x + z = 6 \\2x - 2y + z = 3\(\end{cases}$

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Textbook Question

In Exercises 9 - 16, find the following matrices: a. A + B

Textbook Question

Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

$\begin{bmatrix}\)5 & 0 & 3 & $\vert$ & -11 \\0 & 1 & -4 & $\vert$ & 12 \\7 & 2 & 0 & $\vert$ & 3\(\end{bmatrix}$

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Textbook Question

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B

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