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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblem 19
Chapter 7, Problem 19

Let A=[−3−72−950]A = \(\begin{bmatrix}\)-3 & -7 \\2 & -9 \\5 & 0\(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\)-5 & -1 \\0 & 0 \\3 & -4\(\end{bmatrix}\). Solve each matrix equation for X. 2X + A = B

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Identify the given matrices and the equation to solve: \(2X + A = B\), where \(A = \begin{bmatrix} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{bmatrix}\).
Isolate the term with \(X\) by subtracting matrix \(A\) from both sides of the equation: \(2X = B - A\).
Perform the matrix subtraction \(B - A\) by subtracting corresponding elements of \(A\) from \(B\): \(\left(b_{ij} - a_{ij}\right)\) for each element.
Divide each element of the resulting matrix \$2X\( by 2 to solve for \)X$: \(X = \frac{1}{2}(B - A)\).
Write the final expression for \(X\) as \(X = \frac{1}{2} \left( B - A \right)\), which represents the solution matrix.

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Key Concepts

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

Matrix Addition and Subtraction

Matrix addition and subtraction involve combining corresponding elements from two matrices of the same dimensions. Each element in the resulting matrix is the sum or difference of the elements in the same position from the original matrices. This operation is fundamental for manipulating matrix equations like 2X + A = B.
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Scalar Multiplication of Matrices

Scalar multiplication involves multiplying every element of a matrix by a constant (scalar). For example, multiplying matrix X by 2 means doubling each element of X. This operation is essential for isolating the matrix variable in equations such as 2X + A = B.
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Solving Matrix Equations

To solve matrix equations like 2X + A = B, you isolate the matrix variable by performing inverse operations, such as subtracting A from both sides and then dividing by the scalar. Understanding how to manipulate matrices algebraically is key to finding the unknown matrix X.
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Related Practice
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. -5(A+D)

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Textbook Question

For Exercises 11–22, use Cramer's Rule to solve each system. {2x=3y+25x=51−4y\(\begin{cases}\)2x = 3y + 2 \\5x = 51 - 4y\(\end{cases}\)

Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {x+y−2z=23x−y−6z=−7\(\begin{cases}\)x + y - 2z = 2 \\3x - y - 6z = -7\(\end{cases}\)

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=[−3−72−950]A = \(\begin{bmatrix}\)-3 & -7 \\2 & -9 \\5 & 0\(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\)-5 & -1 \\0 & 0 \\3 & -4\(\end{bmatrix}\). Solve each matrix equation for X. 3X + 2A = B

Textbook Question

For Exercises 11–22, use Cramer's Rule to solve each system. {3x−4y=42x+2y=12\(\begin{cases}\)3x - 4y = 4 \\2x + 2y = 12\(\end{cases}\)