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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 5
Chapter 7, Problem 5

Evaluate each determinant in Exercises 1–10.
∣−7142−4∣\(\begin{vmatrix}\)-7 & 14 \\2 & -4\(\end{vmatrix}\)

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Identify the given 2x2 matrix as \( \begin{bmatrix} -7 & 14 \\ 2 & -4 \end{bmatrix} \).
Recall the formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by \( \text{det} = ad - bc \).
Substitute the values from the matrix into the formula: \( a = -7, b = 14, c = 2, d = -4 \).
Calculate the product of the diagonal elements: \( a \times d = (-7) \times (-4) \).
Calculate the product of the off-diagonal elements: \( b \times c = 14 \times 2 \), then subtract this from the first product to find the determinant.

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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 linear transformations.
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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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Application of Determinants in Algebra

Determinants are used to solve systems of linear equations, find inverses of matrices, and analyze linear transformations. Evaluating determinants is a fundamental skill in college algebra for these applications.
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Related Practice
Textbook Question

Write the augmented matrix for each system of linear equations.

{x−y+z=8y−12z=−15z=1\(\begin{cases}\)x - y + z = 8 \(\y\) - 12z = -15 \(\z\) = 1\(\end{cases}\)

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Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {3x+4y+2z=34x−2y−8z=−4x+y−z=3\(\begin{cases}\)3x + 4y + 2z = 3 \\4x - 2y - 8z = -4 \(\x\) + y - z = 3\(\end{cases}\)

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Textbook Question

In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

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Textbook Question

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. [x4]=[6y]\(\begin{bmatrix}\)x \\4\(\end{bmatrix}\)=\(\begin{bmatrix}\)6 \(\y\]\end{bmatrix}\)

Textbook Question

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

{3x1+5x2−8x3+5x4=−8x1+2x2−3x3+x4=−72x1+3x2−7x3+3x4=−114x1+8x2−10x3+7x4=−10\(\begin{cases}\) 3x_1 + 5x_2 - 8x_3 + 5x_4 = -8 \\ x_1 + 2x_2 - 3x_3 + x_4 = -7 \\ 2x_1 + 3x_2 - 7x_3 + 3x_4 = -11 \\ 4x_1 + 8x_2 - 10x_3 + 7x_4 = -10 \(\end{cases}\)

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Textbook Question

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

A=[−2132−12],B=[1234]A = \(\begin{bmatrix}\) -2 & 1 \\ \(\frac{3}{2}\) & -\(\frac{1}{2}\) \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 1 & 2 \\ 3 & 4 \(\end{bmatrix}\)

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