Find the values of the variables for which each statement is true, if possible. [−7+z4r8s6p25]+[−98r3254]=[2362720712a]\left[ \begin{matrix} -7 + z & 4r & 8s \\ 6p & 2 & 5 \end{matrix} \right] + \left[ \begin{matrix} -9 & 8r & 3 \\ 2 & 5 & 4 \end{matrix} \right] = \left[ \begin{matrix} 2 & 36 & 27 \\ 20 & 7 & 12a \end{matrix} \right]

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 21

6장, 문제 21

Find the values of the variables for which each statement is true, if possible.
$[\begin{matrix} - 7 + z & 4 r & 8 s \\ 6 p & 2 & 5 \end{matrix}] + [\begin{matrix} - 9 & 8 r & 3 \\ 2 & 5 & 4 \end{matrix}] = [\begin{matrix} 2 & 36 & 27 \\ 20 & 7 & 12 a \end{matrix}]$

검증된 단계별 안내

Identify the two given 2x3 matrices on the left side of the equation and the resulting 2x3 matrix on the right side of the equation. Each matrix has 2 rows and 3 columns, and the entries may contain variables.

Recall that matrix addition is performed element-wise, meaning you add corresponding entries from each matrix. For example, if the first entry of the first matrix is $a$ and the first entry of the second matrix is $b$, then the first entry of the sum matrix is $a + b$.

Set up equations by equating each entry of the sum matrix (right side) to the sum of the corresponding entries from the two matrices on the left side. This will give you a system of equations involving the variables.

Solve the system of equations obtained in the previous step. This may involve substitution or elimination methods, depending on the complexity of the variables and equations.

Check your solutions by substituting the variable values back into the original matrices and verifying that their sum equals the given resulting matrix.

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주요 개념

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

Matrix Addition

Matrix addition involves adding corresponding elements from two matrices of the same dimensions. For two matrices A and B, both must have the same number of rows and columns to be added. The resulting matrix has the same dimensions, with each element equal to the sum of the elements in A and B at that position.

Matrix Dimensions

The dimension of a matrix is given by the number of rows and columns it contains, denoted as m x n. Operations like addition and subtraction require matrices to have identical dimensions. Understanding matrix dimensions ensures that operations are valid and helps identify if a solution is possible.

Solving Matrix Equations

Solving matrix equations involves finding unknown variables within matrices that satisfy the given equality. This typically requires equating corresponding elements and solving the resulting system of equations. Mastery of algebraic manipulation and substitution is essential for finding variable values.