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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 23
Chapter 6, Problem 23

Evaluate each determinant.
∣120−12−1014∣\(\left\)| \(\begin{matrix}\) 1 & 2 & 0 \\ -1 & 2 & -1 \\ 0 & 1 & 4 \(\end{matrix}\) \(\right\)|

Verified step by step guidance
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Identify the size of the matrix for which you need to evaluate the determinant (e.g., 2x2, 3x3, etc.). The method depends on the matrix size.
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 for a 3x3 matrix, select a row or column, then calculate the sum of each element multiplied by its cofactor: \(\det = a \cdot C_{11} + b \cdot C_{12} + c \cdot C_{13}\), where each cofactor \(C_{ij} = (-1)^{i+j}\) times the determinant of the minor matrix obtained by removing row \(i\) and column \(j\).
Calculate each minor determinant and apply the signs accordingly, then sum all the terms to find the determinant of the matrix.

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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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Determinants of 2×2 Matrices

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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Determinants of 3×3 Matrices

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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Determinants of 2×2 Matrices
Related Practice
Textbook Question

Solve each system, using the method indicated.

5x + 2y = -10

3x - 5y = -6 (Gauss-Jordan)

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Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (-3)/(x2(x2 + 5))

Textbook Question

Find the inverse, if it exists, for each matrix.[1010−10211]\(\left\)[ \(\begin{matrix}\) 1 & 0 & 1 \\ 0 & -1 & 0 \\ 2 & 1 & 1 \(\end{matrix}\) \(\right\)]

Textbook Question

Graph each inequality. y > (x - 1)2 + 2

Textbook Question

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

x2+y2=102x^2+y^2=102

x2−y2=17x^2-y^2=17

Textbook Question

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

3x + 2y = -9

2x - 5y = -6

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