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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 17
Chapter 6, Problem 17

Find the values of the variables for which each statement is true, if possible.
[xyz]=[216]\(\left\)[ \(\begin{matrix}\) x & y & z \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) 21 & 6 \(\end{matrix}\) \(\right\)]

Verified step by step guidance
1
First, understand that two matrices are equal only if they have the same dimensions (same number of rows and columns) and their corresponding entries are equal.
Check the dimensions of the given matrices: one is a 1x3 matrix (1 row, 3 columns) and the other is a 1x2 matrix (1 row, 2 columns).
Since the matrices have different numbers of columns, they cannot be equal regardless of the values of the variables.
Therefore, conclude that there are no values of the variables that make a 1x3 matrix equal to a 1x2 matrix.
This illustrates the important principle that matrix equality requires matching dimensions before comparing individual entries.

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Key Concepts

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

Matrix Dimensions and Equality

Two matrices are equal only if they have the same dimensions and their corresponding entries are equal. A 1x3 matrix cannot be equal to a 1x2 matrix because their sizes differ, making equality impossible.
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Introduction to Matrices

Matrix Entry Comparison

When matrices have the same dimensions, equality means each element in one matrix equals the corresponding element in the other. This allows setting up equations to solve for variables in the entries.
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Determinants of 3×3 Matrices

Solving Systems of Equations

If matrix entries contain variables, equating corresponding elements forms a system of equations. Solving this system finds the variable values that satisfy the matrix equality.
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Solving Systems of Equations - Substitution
Related Practice
Textbook Question

Find the cofactor of each element in the second row of each matrix.

[12−123−2−141]\(\left\)[ \(\begin{matrix}\) 1 & 2 & -1 \\ 2 & 3 & -2 \\ -1 & 4 & 1 \(\end{matrix}\) \(\right\)]

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

y=x2+6x+9y=x^2+6x+9

x+2y=−2x+2y=-2

Textbook Question

Find the cofactor of each element in the second row of each matrix.

[−201120421]\(\left\)[ \(\begin{matrix}\) -2 & 0 & 1 \\ 1 & 2 & 0 \\ 4 & 2 & 1 \(\end{matrix}\) \(\right\)]

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Textbook Question

Graph each inequality. 2x > 3 - 4y

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Textbook Question

Find the inverse, if it exists, for each matrix. [−12−2−1]\(\left\)[ \(\begin{matrix}\) -1 & 2 \\ -2 & -1 \(\end{matrix}\) \(\right\)]

Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (2x + 1)/(x + 2)^3

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