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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 7
Chapter 7, Problem 7

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. [x2yz9]=[41239]\(\begin{bmatrix}\)x & 2y \(\z\) & 9\(\end{bmatrix}\)=\(\begin{bmatrix}\)4 & 12 \\3 & 9\(\end{bmatrix}\)

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1
Understand that two matrices are equal if and only if their corresponding entries are equal.
Set up equations by equating each corresponding element from the two matrices: \(x = 4\), \(2y = 12\), \(z = 3\), and \(9 = 9\).
Solve the equation \(2y = 12\) by dividing both sides by 2 to find the value of \(y\).
Use the values found for \(x\), \(y\), and \(z\) as the solution to the problem.
Verify your solution by substituting the values back into the original matrices to ensure both matrices are equal.

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Key Concepts

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

Matrix Equality

Two matrices are equal if and only if they have the same dimensions and their corresponding entries are equal. This means each element in one matrix must match the element in the same position in the other matrix.
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Introduction to Matrices

Solving Systems of Equations

When matrices are equal, equate corresponding elements to form equations. Solving these equations simultaneously helps find the values of variables involved, such as x, y, and z in this problem.
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Solving Systems of Equations - Substitution

Substitution and Simplification

After setting up equations from matrix equality, use substitution or algebraic manipulation to isolate variables. Simplifying these equations step-by-step leads to the solution for each variable.
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Related Practice
Textbook Question

Write the augmented matrix for each system of linear equations.

{5x−2y−3z=0x+y=52x−3z=4\(\begin{cases}\)5x - 2y - 3z = 0 \(\x\) + y = 5 \\2x - 3z = 4\(\end{cases}\)

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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. {3x+4y+2z=34x−2y−8z=−4x+y−z=3\(\begin{cases}\)3x + 4y + 2z = 3 \\4x - 2y - 8z = -4 \(\x\) + y - z = 3\(\end{cases}\)

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Textbook Question

Write the augmented matrix for each system of linear equations.

{2w+5x−3y+z=23x+y=4w−x+5y=95w−5x−2y=1\(\begin{cases}\)2w + 5x - 3y + z = 2 \\3x + y = 4 \(\w\) - x + 5y = 9 \\5w - 5x - 2y = 1\(\end{cases}\)

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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. {8x+5y+11z=30−x−4y+2z=32x−y+5z=12\(\begin{cases}\)8x + 5y + 11z = 30 \\-x - 4y + 2z = 3 \\2x - y + 5z = 12\(\end{cases}\)

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Textbook Question

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

A=[010001100],B=[001100010]A = \(\begin{bmatrix}\) 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \(\end{bmatrix}\)

Textbook Question

Evaluate each determinant in Exercises 1–10.

∣−5−1−2−7∣\(\begin{vmatrix}\)-5 & -1 \\-2 & -7\(\end{vmatrix}\)