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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

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

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 sum of each element multiplied by its cofactor: \(\det = \sum a_{ij} C_{ij}\), where \(C_{ij} = (-1)^{i+j} M_{ij}\) and \(M_{ij}\) is the minor determinant.
Compute each minor determinant by removing the row and column of the element, then multiply by the corresponding sign and element, and finally sum all these products to get the determinant value.

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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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Properties of Determinants

Determinants have specific 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 simplify calculations and help verify results when evaluating determinants.
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Methods for Evaluating Determinants

Determinants can be evaluated using various methods depending on matrix size, including expansion by minors and cofactors for larger matrices, and direct formulas for 2x2 or 3x3 matrices. Understanding these methods allows efficient and accurate computation of determinants.
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Related Practice
Textbook Question

Solve each system by substitution.

4x + 3y = -13

-x + y = 5

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Textbook Question

What is the value of x if ∣x00x∣=9\(\begin{vmatrix}\)x & 0\\ 0 & x\(\end{vmatrix}\)=9?

Textbook Question

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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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

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. If we want to solve the following nonlinear system by substitution and we decide to solve equation (2) for y, what will be the resulting equation when the substitution is made into equation (1)?

x2 + y = 2 (1)

x - y = 0 (2)

Textbook Question

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

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