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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 35
Chapter 6, Problem 35

Evaluate each determinant.
∣−1829∣\(\left\)| \(\begin{matrix}\) -1 & 8 \\ 2 & 9 \(\end{matrix}\) \(\right\)|

Verified step by step guidance
1
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, pick a row or column, multiply each element by its corresponding cofactor, and sum the results: \(\det = a(ei - fh) - b(di - fg) + c(dh - eg)\).
After setting up the determinant expression, simplify the arithmetic step-by-step to find 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. It is calculated using specific rules depending on the matrix size, like the product of diagonal elements for 2x2 matrices or expansion by minors for larger matrices.
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Determinants of 2×2 Matrices

Properties of Determinants

Determinants have properties that simplify calculations, such as the determinant of a product equals the product of determinants, and swapping two rows changes the sign of the determinant. Understanding these properties helps in efficiently evaluating determinants without full expansion.
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Methods for Evaluating Determinants

Common methods include expansion by minors (cofactor expansion), row reduction to upper triangular form, and leveraging special matrix types (like diagonal or triangular matrices). Choosing the right method depends on matrix size and structure to simplify the evaluation process.
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Related Practice
Textbook Question

Solve each system of equations. State whether it is an inconsistent system or has infinitely many solutions. If a system has infinitely many solutions, write the solution set with x arbitrary.

5x - 5y - 3 = 0

x - y - 12 = 0

Textbook Question

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

5x2−2y2=255x^2 - 2y^2 = 25

10x2+y2=5010x^2 + y^2 = 50

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Textbook Question

Find each sum or difference, if possible.

[−43]−[582]\(\left\)[ \(\begin{matrix}\) -4 & 3 \(\end{matrix}\) \(\right\)] - \(\left\)[ \(\begin{matrix}\) 5 & 8 & 2 \(\end{matrix}\) \(\right\)]

Textbook Question

Match each inequality with the appropriate calculator graph in A–D. Do not use a calculator y ≤ -3x - 6

Textbook Question

Find each sum or difference, if possible.

[3−42−5−88]−[239−25−732]\(\left\)[ \(\begin{matrix}\) \(\sqrt{3}\) & -4 \\ 2 & -\(\sqrt{5}\) \\ -8 & \(\sqrt{8}\) \(\end{matrix}\) \(\right\)] - \(\left\)[ \(\begin{matrix}\) 2\(\sqrt{3}\) & 9 \\ -2 & \(\sqrt{5}\) \\ -7 & 3\(\sqrt{2}\) \(\end{matrix}\) \(\right\)]

Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x2)/(x4 - 1)