Ch. 6 - Matrices and Determinants

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

Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 15a

Chapter 7, Problem 15a

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

Verified step by step guidance

Step 1: Understand that to find the sum of two matrices A and B, both matrices must have the same dimensions. Here, both A and B are 3x3 matrices, so addition is possible.

Step 2: Recall that matrix addition is performed by adding corresponding elements from each matrix. That is, if A = [a_ij] and B = [b_ij], then A + B = [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: Add each corresponding element: - For the element in the first row, first column: 2 + 6 - For the element in the first row, second column: -10 + 10 - Continue this process for all elements in the matrices.

Step 5: After adding all corresponding elements, write the resulting matrix as the sum A + B.

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Key Concepts

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

Matrix Addition

Matrix addition involves adding corresponding elements from two matrices of the same dimensions. Each element in the resulting matrix is the sum of elements in the same position from the original matrices. This operation is only defined when both matrices have the same number of rows and columns.

Matrix Dimensions

Matrix dimensions are expressed as rows × columns and determine the size of a matrix. For matrix addition, both matrices must have identical dimensions to ensure each element pairs correctly. Understanding dimensions helps verify if operations like addition or multiplication are valid.

Element-wise Operations

Element-wise operations apply a mathematical operation to each corresponding element of matrices. In addition, subtraction, or scalar multiplication, each element is treated independently. This concept is fundamental for performing matrix addition accurately.

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Related Practice

Textbook Question

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

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

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Textbook Question

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

Textbook Question

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

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