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]

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 17

6장, 문제 17

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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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 \(\end{matrix}\) \(\right\)]