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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 5
Chapter 7, Problem 5

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. [x4]=[6y]\(\begin{bmatrix}\)x \\4\(\end{bmatrix}\)=\(\begin{bmatrix}\)6 \(\y\]\end{bmatrix}\)

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Understand that two matrices are equal if and only if their corresponding entries are equal.
Set the corresponding elements of the matrices equal to each other: \(x = 6\) and \(4 = y\).
From the first equation, identify the value of \(x\) as 6.
From the second equation, identify the value of \(y\) as 4.
Conclude that the values of the variables that make the matrices equal are \(x = 6\) and \(y = 4\).

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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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Solving for Variables in Matrices

When matrices contain variables, you can find their values by setting corresponding elements equal to each other. This creates simple equations that can be solved individually to determine the unknown variables.
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Basic Algebraic Manipulation

Solving for variables often requires basic algebraic skills such as isolating the variable, performing inverse operations, and simplifying expressions. These skills are essential to find the values that satisfy the matrix equality.
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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

In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

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Textbook Question

Evaluate each determinant in Exercises 1–10.

∣−7142−4∣\(\begin{vmatrix}\)-7 & 14 \\2 & -4\(\end{vmatrix}\)

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Textbook Question

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

{3x1+5x2−8x3+5x4=−8x1+2x2−3x3+x4=−72x1+3x2−7x3+3x4=−114x1+8x2−10x3+7x4=−10\(\begin{cases}\) 3x_1 + 5x_2 - 8x_3 + 5x_4 = -8 \\ x_1 + 2x_2 - 3x_3 + x_4 = -7 \\ 2x_1 + 3x_2 - 7x_3 + 3x_4 = -11 \\ 4x_1 + 8x_2 - 10x_3 + 7x_4 = -10 \(\end{cases}\)

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Textbook Question

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

A=[−2132−12],B=[1234]A = \(\begin{bmatrix}\) -2 & 1 \\ \(\frac{3}{2}\) & -\(\frac{1}{2}\) \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 1 & 2 \\ 3 & 4 \(\end{bmatrix}\)

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