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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 7
Chapter 6, Problem 7

Use the given row transformation to change each matrix as indicated.
[1447],−4×row 1 added to row 2\(\left\)[ \(\begin{matrix}\) 1 & 4 \\ 4 & 7 \(\end{matrix}\) \(\right\)], \(\quad\) -4 \(\times\) \(\text{row 1 added to row 2}\)

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Identify the given matrix as a 2x2 matrix with elements arranged as: \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), where the first row is \((a, b)\) and the second row is \((c, d)\).
Understand the row operation: '-4 times row 1 added to row 2' means you multiply each element of the first row by -4 and then add the result to the corresponding element in the second row.
Calculate the new second row elements using the formula: \(\text{new row 2} = \text{row 2} + (-4) \times \text{row 1}\), which translates to \(\begin{bmatrix} c + (-4) \times a & d + (-4) \times b \end{bmatrix}\).
Keep the first row unchanged, so the first row remains \(\begin{bmatrix} a & b \end{bmatrix}\).
Write the resulting matrix after the row operation as \(\begin{bmatrix} a & b \\ c - 4a & d - 4b \end{bmatrix}\).

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Key Concepts

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

Matrix Representation and Notation

A matrix is a rectangular array of numbers arranged in rows and columns. Understanding how to read and write matrices, especially 2x2 matrices, is essential for performing operations like row transformations. Each element is identified by its row and column position.
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Interval Notation

Elementary Row Operations

Elementary row operations include swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another. These operations are used to simplify matrices or solve systems of equations. In this problem, adding -4 times row 1 to row 2 modifies the second row accordingly.
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Performing Row Operations on Matrices

Performing Row Transformations

To perform a row transformation, multiply each element of the specified row by the given scalar and add the result to the corresponding element of the target row. This changes the target row while leaving others unchanged, helping to achieve matrix simplification or row echelon form.
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Performing Row Operations on Matrices
Related Practice
Textbook Question

Solve each system by substitution.

4x + 3y = -13

-x + y = 5

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Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. 5/(3x(2x + 1))

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Textbook Question

Evaluate each determinant.

∣−594−1∣\(\left\)| \(\begin{matrix}\) -5 & 9 \\ 4 & -1 \(\end{matrix}\) \(\right\)|

Textbook Question

Use the given row transformation to change each matrix as indicated.

[1−470],−7×row 1 added to row 2\(\left\)[ \(\begin{matrix}\) 1 & -4 \\ 7 & 0 \(\end{matrix}\) \(\right\)], \(\quad\) -7 \(\times\) \(\text{row 1 added to row 2}\)

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Textbook Question

If the dimension of matrix A is 3 ×\(\times\) 2 and the dimension of matrix B is 2 ×\(\times\) 6, then the dimension of AB is ____.

Textbook Question

Find the dimension of each matrix. Identify any square, column, or row matrices.

[−4823]\(\left\)[ \(\begin{matrix}\) -4 & 8 \\ 2 & 3 \(\end{matrix}\) \(\right\)]