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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 21
Chapter 7, Problem 21

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

Verified step by step guidance
1
Start with the given matrix equation: \(3X + 2A = B\).
Isolate the term with \(X\) by subtracting \$2A$ from both sides: \(3X = B - 2A\).
To solve for \(X\), divide both sides of the equation by 3, which is equivalent to multiplying by \(\frac{1}{3}\): \(X = \frac{1}{3}(B - 2A)\).
Calculate the matrix \$2A\( by multiplying each element of matrix \)A$ by 2.
Subtract the matrix \$2A\( from matrix \)B\( element-wise, then multiply the resulting matrix by \(\frac{1}{3}\) to find matrix \)X$.

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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 of two matrices of the same size. Scalar multiplication means multiplying every element of a matrix by a constant. These operations are essential to manipulate the equation 3X + 2A = B by distributing scalars and combining matrices.
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Solving Matrix Equations

To solve matrix equations like 3X + 2A = B, isolate the matrix variable X by performing inverse operations. This typically involves subtracting 2A from both sides and then multiplying by the inverse of the scalar coefficient (here, dividing by 3) to find X.
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Matrix Dimensions and Compatibility

Matrix operations require matrices to have compatible dimensions. Both A and B are 3x2 matrices, so X must also be 3x2 for the equation to be valid. Understanding dimensions ensures correct addition and scalar multiplication without errors.
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Related Practice
Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {w+x−y+z=−22w−x+2y−z=7−w+2x+y+2z=−1\(\begin{cases}\) w + x - y + z = -2 \\ 2w - x + 2y - z = 7 \\ -w + 2x + y + 2z = -1 \(\end{cases}\)

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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. -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 \(\end{cases}\)

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Textbook Question

In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣30021−525−1∣\(\begin{vmatrix}\) 3 & 0 & 0 \\ 2 & 1 & -5 \\ 2 & 5 & -1 \(\end{vmatrix}\)

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Textbook Question

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

{x+y−z=−22x−y+z=5−x+2y+2z=1\(\begin{cases}\) x + y - z = -2 \\ 2x - y + z = 5 \\ -x + 2y + 2z = 1 \(\end{cases}\)

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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. 2X + A = B

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