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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 9
Chapter 7, Problem 9

Evaluate each determinant in Exercises 1–10.∣121218−34∣\(\begin{vmatrix}\[\frac{1}{2}\) & \(\frac{1}{2}\) \(\frac{1}{8}\) & - \(\frac{3}{4}\]\end{vmatrix}\)

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Identify the determinant of the 2x2 matrix given as: \(\begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{8} & -\frac{3}{4} \end{pmatrix}\).
Recall the formula for the determinant of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is \(ad - bc\).
Assign the values: \(a = \frac{1}{2}\), \(b = \frac{1}{2}\), \(c = \frac{1}{8}\), and \(d = -\frac{3}{4}\).
Calculate the product \(ad = \left(\frac{1}{2}\right) \times \left(-\frac{3}{4}\right)\) and the product \(bc = \left(\frac{1}{2}\right) \times \left(\frac{1}{8}\right)\).
Subtract the two products to find the determinant: \(ad - bc\).

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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 of the matrix and is essential in solving systems of linear equations.
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Fraction Arithmetic

When working with fractions in matrices, it is important to correctly perform multiplication and subtraction. Multiplying fractions involves multiplying numerators and denominators, while subtraction requires a common denominator.
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Simplification of Expressions

After calculating the determinant expression, simplifying the resulting fraction or mixed number is necessary. This involves reducing fractions to their simplest form to provide a clear and final answer.
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Related Practice
Textbook Question

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

A=[123134143],B=[72−312−12012−121−12]A = \(\begin{bmatrix}\) 1 & 2 & 3 \\ 1 & 3 & 4 \\ 1 & 4 & 3 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) \(\frac{7}{2}\) & -3 & \(\frac{1}{2}\) \\ -\(\frac{1}{2}\) & 0 & \(\frac{1}{2}\) \\ -\(\frac{1}{2}\) & 1 & -\(\frac{1}{2}\) \(\end{bmatrix}\)

Textbook Question

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

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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. {w−2x−y−3z=−9w+x−y=03w+4x+z=62x−2y+z=3\(\begin{cases}\)w - 2x - y - 3z = -9 \(\w\) + x - y = 0 \\3w + 4x + z = 6 \\2x - 2y + z = 3\(\end{cases}\)

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Textbook Question

In Exercises 9 - 16, find the following matrices: a. A + B

Textbook Question

Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

[503∣−1101−4∣12720∣3]\(\begin{bmatrix}\)5 & 0 & 3 & \(\vert\) & -11 \\0 & 1 & -4 & \(\vert\) & 12 \\7 & 2 & 0 & \(\vert\) & 3\(\end{bmatrix}\)

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Textbook Question

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