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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 57
Capitolo 7, Problema 57

In Exercises 57–60, solve each equation for x.

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Step 1: Recognize that the problem involves finding the value of \( x \) such that the determinant of the 2x2 matrix \( \begin{bmatrix} -2 & x \\ 4 & 6 \end{bmatrix} \) equals 32.
Step 2: Recall the formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by \( ad - bc \).
Step 3: Apply the determinant formula to the given matrix: \( (-2)(6) - (x)(4) = 32 \).
Step 4: Simplify the expression: \( -12 - 4x = 32 \).
Step 5: Solve the resulting linear equation for \( x \) by isolating \( x \) on one side.

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Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This scalar value provides important properties of the matrix, such as invertibility. In this problem, the determinant is set equal to 32, which forms an equation to solve for x.
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Determinants of 2×2 Matrices

Setting up and Solving Algebraic Equations

After expressing the determinant as an algebraic expression, you form an equation equal to 32. Solving this equation involves isolating the variable x using algebraic operations like addition, subtraction, multiplication, and division to find its value.
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Solving Logarithmic Equations

Properties of Absolute Value and Determinants

The vertical bars around the matrix indicate the determinant, not absolute value of a number. Understanding this notation is crucial to avoid confusion and correctly interpret the problem as finding the determinant rather than an absolute value.
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Determinants of 2×2 Matrices
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Use Cramer's Rule to solve each system.

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In Exercises 55–56, write the system of linear equations for which Cramer's Rule yields the given determinants.

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In Exercises 57–60, solve each equation for x.

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Solve the system

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