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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 21
Chapter 6, Problem 21

Evaluate each determinant.
∣4−78213−630∣\(\left\)| \(\begin{matrix}\) 4 & -7 & 8 \\ 2 & 1 & 3 \\ -6 & 3 & 0 \(\end{matrix}\) \(\right\)|

Verified step by step guidance
1
Identify the size of the determinant matrix (e.g., 2x2, 3x3) to determine the appropriate method for evaluation.
For a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), use the formula for the determinant: \(\det = ad - bc\).
For a 3x3 matrix \(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\), apply the rule of Sarrus or cofactor expansion to find the determinant.
If using cofactor expansion, select a row or column (usually one with zeros for simplicity), then calculate the minors and cofactors for each element in that row or column.
Sum the products of each element and its corresponding cofactor to get the determinant value: \(\det = a_{ij}C_{ij} + a_{ik}C_{ik} + \dots\) where \(C_{ij}\) are cofactors.

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Key Concepts

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

Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that provides important properties about the matrix, such as invertibility. For a 2x2 matrix, it is calculated as ad - bc, where a, b, c, and d are the elements of the matrix. Determinants help in solving systems of linear equations and understanding matrix behavior.
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Methods for Calculating Determinants

Determinants can be calculated using various methods depending on the matrix size, including expansion by minors, cofactor expansion, and row reduction. For larger matrices, breaking down into smaller matrices or using properties like linearity simplifies the process. Understanding these methods is essential for efficient evaluation.
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Properties of Determinants

Determinants have key properties such as changing sign when two rows are swapped, being zero if rows are linearly dependent, and the determinant of a product equals the product of determinants. These properties aid in simplifying calculations and interpreting the results in algebraic contexts.
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Related Practice
Textbook Question

Find the cofactor of each element in the second row of each matrix.

[12−123−2−141]\(\left\)[ \(\begin{matrix}\) 1 & 2 & -1 \\ 2 & 3 & -2 \\ -1 & 4 & 1 \(\end{matrix}\) \(\right\)]

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

y=6x+x2y=6x+x^2

4x−y=−34x-y=-3

Textbook Question

Find the values of the variables for which each statement is true, if possible.

[05x−13y+241z]=[0w6−130418]\(\left\)[ \(\begin{matrix}\) 0 & 5 & x \\ -1 & 3 & y + 2 \\ 4 & 1 & z \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) 0 & w & 6 \\ -1 & 3 & 0 \\ 4 & 1 & 8 \(\end{matrix}\) \(\right\)]

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Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x3 + 4)/(9x3 - 4x)

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Textbook Question

Find the values of the variables for which each statement is true, if possible.

[−7+z4r8s6p25]+[−98r3254]=[2362720712a]\(\left\)[ \(\begin{matrix}\) -7 + z & 4r & 8s \\ 6p & 2 & 5 \(\end{matrix}\) \(\right\)] + \(\left\)[ \(\begin{matrix}\) -9 & 8r & 3 \\ 2 & 5 & 4 \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) 2 & 36 & 27 \\ 20 & 7 & 12a \(\end{matrix}\) \(\right\)]

Textbook Question

Graph each inequality. y < 3x2 + 2

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