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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblem 7
Chapter 7, Problem 7

Evaluate each determinant in Exercises 1–10.
∣−5−1−2−7∣\(\begin{vmatrix}\)-5 & -1 \\-2 & -7\(\end{vmatrix}\)

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1
Identify the matrix given as a 2x2 matrix: \(\begin{bmatrix} -5 & -1 \\ -2 & -7 \end{bmatrix}\).
Recall the formula for the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), which is \(ad - bc\).
Assign the values from the matrix to the variables: \(a = -5\), \(b = -1\), \(c = -2\), and \(d = -7\).
Substitute these values into the determinant formula: \((-5)(-7) - (-1)(-2)\).
Simplify the expression 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 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This scalar value helps determine properties like invertibility and area scaling in transformations.
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Determinants of 2×2 Matrices

Matrix Notation and Elements

Understanding matrix notation involves recognizing the position of elements: 'a' and 'b' in the first row, 'c' and 'd' in the second. Correctly identifying these values is essential for accurate determinant calculation.
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Interval Notation

Application of Determinants in Problem Solving

Evaluating determinants is a fundamental skill in algebra used to solve systems of equations, find inverses, and analyze linear transformations. Mastery of this concept aids in broader mathematical problem solving.
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Related Practice
Textbook Question

Write the augmented matrix for each system of linear equations.

{5x−2y−3z=0x+y=52x−3z=4\(\begin{cases}\)5x - 2y - 3z = 0 \(\x\) + y = 5 \\2x - 3z = 4\(\end{cases}\)

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Textbook Question

Write the augmented matrix for each system of linear equations.

{2w+5x−3y+z=23x+y=4w−x+5y=95w−5x−2y=1\(\begin{cases}\)2w + 5x - 3y + z = 2 \\3x + y = 4 \(\w\) - x + 5y = 9 \\5w - 5x - 2y = 1\(\end{cases}\)

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Textbook Question

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. [x2yz9]=[41239]\(\begin{bmatrix}\)x & 2y \(\z\) & 9\(\end{bmatrix}\)=\(\begin{bmatrix}\)4 & 12 \\3 & 9\(\end{bmatrix}\)

Textbook Question

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {8x+5y+11z=30−x−4y+2z=32x−y+5z=12\(\begin{cases}\)8x + 5y + 11z = 30 \\-x - 4y + 2z = 3 \\2x - y + 5z = 12\(\end{cases}\)

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Textbook Question

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

A=[010001100],B=[001100010]A = \(\begin{bmatrix}\) 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \(\end{bmatrix}\)