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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 3
Chapter 7, Problem 3

Evaluate each determinant in Exercises 1–10.
∣−4156∣\(\begin{vmatrix}\)-4 & 1 \\5 & 6\(\end{vmatrix}\)

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1
Identify the matrix given as \( \begin{pmatrix} -4 & 1 \\ 5 & 6 \end{pmatrix} \).
Recall the formula for the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by \( \text{det} = ad - bc \).
Substitute the values from the matrix into the formula: \( a = -4, b = 1, c = 5, d = 6 \).
Calculate the product of the diagonal elements: \( a \times d = (-4) \times 6 \).
Calculate the product of the off-diagonal elements: \( b \times c = 1 \times 5 \), then subtract this from the previous 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' are 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

Applications of Determinants

Determinants are used to solve systems of linear equations, find matrix inverses, and analyze geometric transformations. Evaluating determinants is a foundational skill in linear algebra and college algebra.
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Related Practice
Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {5x+8y−6z=143x+4y−2z=8x+2y−2z=3\(\begin{cases}\)5x + 8y - 6z = 14 \\3x + 4y - 2z = 8 \(\x\) + 2y - 2z = 3\(\end{cases}\)

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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

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

A=[−4013],B=[−2401]A = \(\begin{bmatrix}\) -4 & 0 \\ 1 & 3 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) -2 & 4 \\ 0 & 1 \(\end{bmatrix}\)

Textbook Question

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

{x+2y+3z=−52x+y+z=1x+y−z=8\(\begin{cases}\) x + 2y + 3z = -5 \\ 2x + y + z = 1 \\ x + y - z = 8 \(\end{cases}\)

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Textbook Question

Evaluate each determinant in Exercises 1–10.

∣5723∣\(\begin{vmatrix}\)5 & 7 \\2 & 3\(\end{vmatrix}\)

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Textbook Question

a. Give the order of each matrix.


b. If A = [aᵢⱼ] , identify a₃₂ and a₂₃, or explain why identification is not possible.

[1−5πe07−6−π−21211−15]\(\begin{bmatrix}\)1 & -5 & \(\pi\) & e \\0 & 7 & -6 & -\(\pi\) \\-2 & \(\frac{1}{2}\) & 11 & -\(\frac{1}{5}\]\end{bmatrix}\)