Find each sum or difference, if possible. [3−42−5−88]−[239−25−732]\left[ \begin{matrix} \sqrt{3} & -4 \\ 2 & -\sqrt{5} \\ -8 & \sqrt{8} \end{matrix} \right] - \left[ \begin{matrix} 2\sqrt{3} & 9 \\ -2 & \sqrt{5} \\ -7 & 3\sqrt{2} \end{matrix} \right]

Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 37

Chapter 6, Problem 37

Find each sum or difference, if possible.
$[\begin{matrix} \sqrt{3} & - 4 \\ 2 & - \sqrt{5} \\ - 8 & \sqrt{8} \end{matrix}] - [\begin{matrix} 2 \sqrt{3} & 9 \\ - 2 & \sqrt{5} \\ - 7 & 3 \sqrt{2} \end{matrix}]$

Verified step by step guidance

Identify the dimensions of both matrices to ensure they are the same. Since both are 3x2 matrices, addition or subtraction is possible.

Recall that to add or subtract matrices, you perform the operation element-wise. This means subtracting corresponding elements from each matrix.

Label the elements of the first matrix as $a_{ij}$ and the second matrix as $b_{ij}$, where $i$ is the row number and $j$ is the column number.

Form the resulting matrix by calculating each element as $c_{ij} = a_{ij} - b_{ij}$ for all $i$ and $j$.

Write the resulting 3x2 matrix with the new elements $c_{ij}$, which represents the difference of the two matrices.

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Key Concepts

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

Matrix Dimensions and Compatibility

Matrix addition and subtraction require the 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. In this question, both matrices are 3x2, so they are compatible for addition or subtraction.

Matrix Addition and Subtraction

To add or subtract matrices, perform the operation element-wise by adding or subtracting corresponding entries from each matrix. For example, subtract the element in row 1, column 1 of the second matrix from the element in the same position of the first matrix, and repeat for all elements.

Representation and Notation of Matrices

A matrix is a rectangular array of numbers arranged in rows and columns, denoted by brackets. Understanding how to read and write matrices, including identifying elements by their row and column positions, is essential for performing operations like addition and subtraction.

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Textbook Question

Evaluate each determinant.

$∣ \begin{matrix} x & 4 x \\ 2 x & 8 x \end{matrix} ∣$