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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 77
Chapter 6, Problem 77

Perform each operation, if possible.
[−1025][−3428]\(\left\)[ \(\begin{matrix}\) -1 & 0 \\ 2 & 5 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) -3 & 4 \\ 2 & 8 \(\end{matrix}\) \(\right\)]

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Identify the two 2x2 matrices given in the problem. Let's call the first matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) and the second matrix \( B = \begin{bmatrix} e & f \\ g & h \end{bmatrix} \).
Determine the operation to perform between the two matrices. Common operations include addition, subtraction, and multiplication. Confirm which operation is requested.
If the operation is addition or subtraction, add or subtract the corresponding elements of the matrices: \( A \pm B = \begin{bmatrix} a \pm e & b \pm f \\ c \pm g & d \pm h \end{bmatrix} \).
If the operation is multiplication, multiply the matrices using the rule for matrix multiplication: \( (AB)_{ij} = \sum_{k=1}^2 A_{ik} B_{kj} \). Specifically, calculate each element of the product matrix as follows:
\( \begin{bmatrix} a \times e + b \times g & a \times f + b \times h \\ c \times e + d \times g & c \times f + d \times h \end{bmatrix} \).

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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

Understanding the size of matrices (rows × columns) is essential to determine if operations like addition, subtraction, or multiplication are possible. For addition and subtraction, matrices must have identical dimensions. For multiplication, the number of columns in the first matrix must equal the number of rows in the second.
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Matrix Addition and Subtraction

Matrix addition and subtraction involve combining corresponding elements from two matrices of the same size. Each element in the resulting matrix is the sum or difference of elements in the same position from the original matrices.
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Matrix Multiplication

Matrix multiplication involves taking the dot product of rows from the first matrix with columns from the second. The resulting matrix has dimensions equal to the number of rows of the first matrix and columns of the second. This operation is not element-wise and requires compatible dimensions.
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Related Practice
Textbook Question

The graphs show regions of feasible solutions. Find the maximum and minimum values of each objective function. objective function = 3x + 5y

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Textbook Question

Consider the following nonlinear system. Work Exercises 75 –80 in order.

y = | x - 1 |

y = x2 - 4

How is the graph of y = | x - 1 | obtained by transforming the graph of y = | x |?

Textbook Question

For what value(s) of k will the following system of linear equations have no solution? infinitely many solutions?

x - 2y = 3

-2x + 4y = k

Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

x + 2y + 3z = 4

4x + 3y + 2z = 1

-x - 2y - 3z = 0

Textbook Question

Use a system of equations to solve each problem. See Example 8. Find an equation of the line y = ax + b that passes through the points (-2, 1) and (-1, -2).

Textbook Question

Consider the following nonlinear system. Work Exercises 75 –80 in order.

y = | x - 1 |

y = x2 - 4

Use the definition of absolute value to write y = | x - 1 | as a piecewise-defined function.

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