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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 81
Chapter 6, Problem 81

Perform each operation, if possible.
[−2550143−4−1][10−1−10011−1]\(\left\)[ \(\begin{matrix}\) -2 & 5 & 5 \\ 0 & 1 & 4 \\ 3 & -4 & -1 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) 1 & 0 & -1 \\ -1 & 0 & 0 \\ 1 & 1 & -1 \(\end{matrix}\) \(\right\)]

Verified step by step guidance
1
Identify the operation to be performed between the two 3x3 matrices. Common operations include addition, subtraction, and multiplication. Confirm which operation is requested.
If the operation is addition or subtraction, verify that both matrices have the same dimensions (which they do, both are 3x3). Then, add or subtract corresponding elements from each matrix. For example, if the first matrix element is \(a_{ij}\) and the second is \(b_{ij}\), the result element is \(c_{ij} = a_{ij} \pm b_{ij}\).
If the operation is multiplication, recall that the product of two 3x3 matrices \(A\) and \(B\) is another 3x3 matrix \(C\), where each element \(c_{ij}\) is calculated by taking the dot product of the \(i\)-th row of \(A\) with the \(j\)-th column of \(B\). Mathematically, \(c_{ij} = \sum_{k=1}^3 a_{ik} b_{kj}\).
To compute each element \(c_{ij}\) in the product matrix, multiply corresponding elements from the \(i\)-th row of the first matrix and the \(j\)-th column of the second matrix, then sum these products. Repeat this for all \(i\) and \(j\) from 1 to 3.
After computing all elements, assemble them into the resulting 3x3 matrix. This matrix is the final result of the operation.

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Key Concepts

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

Matrix Addition and Subtraction

Matrix addition and subtraction involve combining two matrices of the same dimensions by adding or subtracting their corresponding elements. Each element in the resulting matrix is found by performing the operation on elements in the same position from both matrices.
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Matrix Multiplication

Matrix multiplication requires that the number of columns in the first matrix equals the number of rows in the second. The product matrix's elements are calculated by taking the dot product of rows from the first matrix with columns from the second, summing the products of corresponding entries.
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Dimension Compatibility

Operations on matrices are only possible when their dimensions meet specific criteria: addition and subtraction require identical dimensions, while multiplication requires the first matrix's columns to match the second matrix's rows. Understanding these rules ensures the operation is valid.
Related Practice
Textbook Question

For each pair of matrices A and B, find (a) AB and (b) BA.

A=[−101011−1−10],B=[001010100]A = \(\left\)[ \(\begin{matrix}\) -1 & 0 & 1 \\ 0 & 1 & 1 \\ -1 & -1 & 0 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \(\end{matrix}\) \(\right\)]

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Textbook Question

For each pair of matrices A and B, find (a) AB and (b) BA. A=[0−5−42],B=[3−1−54]A = \(\left\)[ \(\begin{matrix}\) 0 & -5 \\ -4 & 2 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 3 & -1 \\ -5 & 4 \(\end{matrix}\) \(\right\)]

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Textbook Question

Find the maximum and minimum values of each objective function over the region of feasible solutions shown at the right. objective function = 10y

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Textbook Question

Find AB and BA for the following matrices.

A=[abcd]andB=[1001]A = \(\left\)[ \(\begin{matrix}\) a & b \\ c & d \(\end{matrix}\) \(\right\)] \(\quad\) \(\text{and}\) \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 1 & 0 \\ 0 & 1 \(\end{matrix}\) \(\right\)]

Matrix B acts as the multiplicative element for 2 ×\(\times\) 2 square matrices.

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Textbook Question

Find the inverse, if it exists, for each matrix.

[2153]\(\left\)[ \(\begin{matrix}\) 2 & 1 \\ 5 & 3 \(\end{matrix}\) \(\right\)]

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Textbook Question

For each pair of matrices A and B, find (a) AB and (b) BA. A=[01−1010001],B=[100010001]A = \(\left\)[ \(\begin{matrix}\) 0 & 1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \(\end{matrix}\) \(\right\)]

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