Find each sum or difference, if possible. [6−92413]+[−8256−34]\left[ \begin{matrix} 6 & -9 & 2 \\ 4 & 1 & 3 \end{matrix} \right] + \left[ \begin{matrix} -8 & 2 & 5 \\ 6 & -3 & 4 \end{matrix} \right]

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 27

6장, 문제 27

Find each sum or difference, if possible.
$[\begin{matrix} 6 & - 9 & 2 \\ 4 & 1 & 3 \end{matrix}] + [\begin{matrix} - 8 & 2 & 5 \\ 6 & - 3 & 4 \end{matrix}]$

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Identify the dimensions of both matrices to ensure they are the same. Since both are 2x3 matrices, addition or subtraction is possible.

Recall that matrix addition (or subtraction) is performed element-wise. This means you add (or subtract) corresponding elements from each matrix.

Write down the general formula for matrix addition: if $A = [a_{ij}]$ and $B = [b_{ij}]$, then $A + B = [a_{ij} + b_{ij}]$ for all $i$ and $j$.

Add the corresponding elements from each matrix. For example, add the element in the first row and first column of the first matrix to the element in the first row and first column of the second matrix, and continue this for all elements.

After performing the element-wise addition, write the resulting matrix with the sums in their respective positions to complete the solution.

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주요 개념

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Matrix Dimensions and Compatibility

Matrix addition or subtraction requires both matrices to have the same dimensions, meaning the same number of rows and columns. If the matrices differ in size, their sum or difference is undefined. Understanding this ensures that operations are only performed on compatible matrices.

Matrix Addition and Subtraction

To add or subtract matrices, you perform the operation element-wise, combining corresponding entries from each matrix. For example, the element in the first row and first column of the result is the sum or difference of the elements in the first row and first column of the original matrices.

Notation and Representation of Matrices

Matrices are typically represented as rectangular arrays enclosed in brackets, with rows and columns clearly defined. Understanding how to read and write matrices correctly is essential for performing operations and interpreting results in algebra.

Find each sum or difference, if possible. [6−92413]+[−8256−34]\(\left\)[ \(\begin{matrix}\) 6 & -9 & 2 \\ 4 & 1 & 3 \(\end{matrix}\) \(\right\)] + \(\left\)[ \(\begin{matrix}\) -8 & 2 & 5 \\ 6 & -3 & 4 \(\end{matrix}\) \(\right\)]