Ch. 6 - Matrices and Determinants

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

Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 15b

Chapter 7, Problem 15b

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, which means subtracting matrix B from matrix A. Both matrices A and B are 3x3 matrices, so subtraction is possible element-wise.

Step 2: Recall the rule for matrix subtraction. For two matrices A = [a_ij] and B = [b_ij] of the same size, the difference A - B is a matrix C = [c_ij] where each element c_ij = a_ij - b_ij.

Step 3: Write down the elements of matrices A and B clearly: A = $\begin{bmatrix}$ 2 & -10 & -2 \\ 14 & 12 & 10 \\ 4 & -2 & 2 $\end{bmatrix}$, B = $\begin{bmatrix}$ 6 & 10 & -2 \\ 0 & -12 & -4 \\ -5 & 2 & -2 $\end{bmatrix}$

Step 4: Subtract corresponding elements of B from A. For example, the element in the first row and first column of A - B is 2 - 6, the element in the first row and second column is -10 - 10, and so on for all elements.

Step 5: Write the resulting matrix after subtraction by placing each computed element in its corresponding position to form the matrix A - B.

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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

For matrix operations like addition or subtraction to be valid, the matrices must have the same dimensions, meaning the same number of rows and columns. Here, both matrices A and B are 3x3, allowing element-wise subtraction.

Element-wise Operations

Element-wise operations apply arithmetic operations to each corresponding element in matrices. In subtraction, this means subtracting each element of matrix B from the corresponding element of matrix A to form the new matrix.

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

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In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. D-A

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Textbook Question

Use the fact that if $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , then $A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ to find the inverse of each matrix, if possible. Check that $A A^{- 1} = I_{2}$ and $A^{- 1} A = I_{2}$ .

$A = $\begin{bmatrix}$3 & -1 \\-4 & 2$\end{bmatrix}$$

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Textbook Question

Perform each matrix row operation and write the new matrix.

$$\begin{bmatrix}$1 & -3 & 2 & $\vert$ & 0 \\3 & 1 & -1 & $\vert$ & 7 \\2 & -2 & 1 & $\vert$ & 3$\end{bmatrix}\]\quad$ -3R_1 + R_2$

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

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Textbook Question

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