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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 1
Chapter 7, Problem 1

Evaluate each determinant in Exercises 1–10.
∣5723∣\(\begin{vmatrix}\)5 & 7 \\2 & 3\(\end{vmatrix}\)

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1
Identify the matrix given as a 2x2 matrix: \(\begin{bmatrix} 5 & 7 \\ 2 & 3 \end{bmatrix}\).
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: \(a = 5\), \(b = 7\), \(c = 2\), and \(d = 3\).
Substitute these values into the determinant formula: \(5 \times 3 - 7 \times 2\).
Simplify the expression to find the determinant value (do not calculate the final number here).

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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. It is a scalar value that provides important information about the matrix, such as whether it is invertible. For the given matrix [[5, 7], [2, 3]], the determinant is (5)(3) - (7)(2).
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Matrix Notation and Elements

A matrix is a rectangular array of numbers arranged in rows and columns. Each element is identified by its row and column position. Understanding how to read and identify elements in a matrix is essential for performing operations like finding determinants.
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Interval Notation

Properties and Applications of Determinants

Determinants help determine if a matrix is invertible (non-zero determinant) and are used in solving systems of linear equations, finding area or volume transformations, and understanding linear transformations. Recognizing these applications helps contextualize why calculating determinants is important.
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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

a. Give the order of each matrix.


b. If A=[aij]A = [a_{ij}], identify a32a_{32} and a23a_{23}, or explain why identification is not possible.

[4−75−68−1]\(\begin{bmatrix}\)4 & -7 & 5 \\-6 & 8 & -1\(\end{bmatrix}\)

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Textbook Question

Evaluate each determinant in Exercises 1–10.

∣−4156∣\(\begin{vmatrix}\)-4 & 1 \\5 & 6\(\end{vmatrix}\)

Textbook Question

Perform each matrix row operation and write the new matrix.

[122∣201−1∣2054∣1]−5R2+R3\(\begin{bmatrix}\)1 & 2 & 2 & \(\vert\) & 2 \\0 & 1 & -1 & \(\vert\) & 2 \\0 & 5 & 4 & \(\vert\) & 1\(\end{bmatrix}\)-5R_2 + R_3

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Textbook Question

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

A=[4−3−54],B=[4354]A = \(\begin{bmatrix}\)4 & -3 \\-5 & 4\(\end{bmatrix}\), B = \(\begin{bmatrix}\)4 & 3 \\5 & 4\(\end{bmatrix}\)

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}\)