Ch. 6 - Matrices and Determinants

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

Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 15d

Chapter 7, Problem 15d

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

Verified step by step guidance

Step 1: Understand the problem. You are asked to find the matrix expression $-3A + 2B$, where $A$ and $B$ are given matrices.

Step 2: Multiply matrix $A$ by the scalar $-3$. This means multiplying every element of matrix $A$ by $-3$. For example, the element in the first row and first column of $A$ is 2, so after multiplication it becomes $-3 \times 2 = -6$. Do this for all elements of $A$.

Step 3: Multiply matrix $B$ by the scalar $2$. This means multiplying every element of matrix $B$ by $2$. For example, the element in the first row and first column of $B$ is 6, so after multiplication it becomes $2 \times 6 = 12$. Do this for all elements of $B$.

Step 4: Add the resulting matrices from Step 2 and Step 3. To add two matrices, add their corresponding elements. For example, add the element in the first row and first column of the scaled $A$ matrix to the element in the first row and first column of the scaled $B$ matrix.

Step 5: Write the resulting matrix from Step 4 as your final answer for $-3A + 2B$.

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Key Concepts

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

Matrix Addition and Scalar Multiplication

Matrix addition involves adding corresponding elements from two matrices of the same dimensions. Scalar multiplication means multiplying every element of a matrix by a constant. These operations are fundamental for combining matrices as in the expression -3A + 2B.

Matrix Dimensions and Compatibility

For matrix addition or subtraction, the matrices must have the same dimensions (same number of rows and columns). Here, both A and B are 3x3 matrices, so operations like -3A + 2B are valid and can be performed element-wise.

Order of Operations in Matrix Expressions

When evaluating expressions like -3A + 2B, scalar multiplication is performed first on each matrix, followed by matrix addition. This ensures the correct combination of matrices and accurate results.

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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}$10 & -2 \\-5 & 1$\end{bmatrix}$$

Textbook Question

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

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases}\)x + 2y + 3z = 5 $\y$ - 5z = 0\(\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: a. A + B