Ch. 6 - Matrices and Determinants

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

Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 20

Chapter 7, Problem 20

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. -5(A+D)

Verified step by step guidance

Step 1: Understand the problem. The problem involves performing matrix operations. You are tasked with calculating -5(A + D), where A and D are matrices. Ensure that the matrices A and D are defined and have the same dimensions, as matrix addition is only defined for matrices of the same size.

Step 2: Add the matrices A and D. To do this, add the corresponding elements of matrices A and D. For example, if A = [[a11, a12], [a21, a22]] and D = [[d11, d12], [d21, d22]], then A + D = [[a11 + d11, a12 + d12], [a21 + d21, a22 + d22]].

Step 3: Multiply the resulting matrix (A + D) by the scalar -5. To perform scalar multiplication, multiply each element of the matrix (A + D) by -5. For example, if A + D = [[x11, x12], [x21, x22]], then -5(A + D) = [[-5 * x11, -5 * x12], [-5 * x21, -5 * x22]].

Step 4: Verify your calculations. Double-check that the addition of matrices A and D was performed correctly and that each element of the resulting matrix was multiplied by -5 accurately.

Step 5: If the dimensions of A and D are not the same, state that the operation is not defined because matrix addition requires matrices to have the same dimensions.

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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 combining two matrices of the same dimensions by adding their corresponding elements. For example, if A and D are both 2x2 matrices, the sum A + D is obtained by adding each element in A to the corresponding element in D. This operation is only defined when the matrices have the same size.

Scalar Multiplication

Scalar multiplication refers to the process of multiplying each element of a matrix by a scalar (a single number). In the expression -5(A + D), the result of the matrix addition A + D is multiplied by -5, which scales each element of the resulting matrix by -5. This operation is defined for any matrix regardless of its dimensions.

Matrix Dimensions

Matrix dimensions describe the size of a matrix in terms of rows and columns, denoted as 'm x n' where m is the number of rows and n is the number of columns. Understanding dimensions is crucial for determining whether matrix operations, such as addition or multiplication, can be performed. If A and D have different dimensions, the operation A + D would be undefined.

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Related Practice

Textbook Question

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases}\)2x = 3y + 2 \\5x = 51 - 4y\(\end{cases}$

Textbook Question

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

$\begin{cases}\)x + y - z = -2 \\2x - y + z = 5 \\-x + 2y + 2z = 1\(\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. 2X + A = B

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Textbook Question

In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s, are shown. Fill in the missing numbers in the steps that are shown.

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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. 3X + 2A = B

Textbook Question

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases}\)3x - 4y = 4 \\2x + 2y = 12\(\end{cases}$