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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 17
Chapter 7, Problem 17

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. X - A = B

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Identify the given matrices: \( A = \begin{bmatrix} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{bmatrix} \). The equation to solve is \( X - A = B \).
To isolate \( X \), add matrix \( A \) to both sides of the equation: \( X - A + A = B + A \), which simplifies to \( X = B + A \).
Perform matrix addition by adding corresponding elements of matrices \( B \) and \( A \). For example, the element in the first row and first column of \( X \) is \( B_{11} + A_{11} = -5 + (-3) \).
Continue adding each corresponding element: \( X_{12} = B_{12} + A_{12} = -1 + (-7) \), \( X_{21} = B_{21} + A_{21} = 0 + 2 \), and so on for all elements.
Write the resulting matrix \( X \) after completing the addition of all corresponding elements from \( A \) and \( B \).

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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. To subtract matrix A from matrix X, each element in A is subtracted from the corresponding element in X. This operation is essential for solving equations like X - A = B.
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Solving Matrix Equations

To solve matrix equations such as X - A = B, isolate the unknown matrix X by performing inverse operations. Here, adding matrix A to both sides yields X = B + A. Understanding how to manipulate matrices algebraically is crucial for finding the solution.
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Matrix Dimensions and Compatibility

Matrix operations require that matrices have compatible dimensions. Both A and B must be the same size to perform addition or subtraction. Recognizing the dimensions ensures valid operations and helps avoid errors when solving matrix equations.
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Introduction to Matrices
Related Practice
Textbook Question

Use the fact that if A=[abcd]A=\(\begin{bmatrix}\)a & b\\ c & d\(\end{bmatrix}\), then A−1=1ad−bc[d−b−ca]A^{-1}=\(\frac{1}{ad-bc}\]\begin{bmatrix}\)d & -b\\ -c & a\(\end{bmatrix}\) to find the inverse of each matrix, if possible. Check that AA−1=I2AA^{-1} = I_2 and A−1A=I2A^{-1}A = I_2.

A=[10−2−51]A = \(\begin{bmatrix}\)10 & -2 \\-5 & 1\(\end{bmatrix}\)

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

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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. 3A+2D

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Textbook Question

For Exercises 11–22, use Cramer's Rule to solve each system. {x+2y=33x−4y=4\(\begin{cases}\)x + 2y = 3 \\3x - 4y = 4\(\end{cases}\)

Textbook Question

Perform each matrix row operation and write the new matrix.

[1−111∣301−2−1∣02034∣115124∣6]−2R1+R3−5R1+R4\(\begin{bmatrix}\)1 & -1 & 1 & 1 & \(\vert\) & 3 \\0 & 1 & -2 & -1 & \(\vert\) & 0 \\2 & 0 & 3 & 4 & \(\vert\) & 11 \\5 & 1 & 2 & 4 & \(\vert\) & 6\(\end{bmatrix}\[\quad\]\begin{array}{l}\)-2R_1 + R_3 \\-5R_1 + R_4\(\end{array}\)

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