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Use Cramer's Rule to solve each system.
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Ch. 6 - Matrices and Determinants
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 7, Problema 59
In Exercises 57–60, solve each equation for x.
Guida verificata passo dopo passo1
Step 1: Recognize that the problem involves finding the value of \( x \) such that the determinant of the given 3x3 matrix equals \( -8 \). The matrix is \( \begin{bmatrix} 1 & x & -2 \\ 3 & 1 & 1 \\ 0 & -2 & 2 \end{bmatrix} \).
Step 2: Recall the formula for the determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) is \( \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).
Step 3: Apply the determinant formula to the given matrix, substituting \( a=1 \), \( b=x \), \( c=-2 \), \( d=3 \), \( e=1 \), \( f=1 \), \( g=0 \), \( h=-2 \), and \( i=2 \).
Step 4: Calculate each minor determinant: \( ei - fh = (1)(2) - (1)(-2) \), \( di - fg = (3)(2) - (1)(0) \), and \( dh - eg = (3)(-2) - (1)(0) \).
Step 5: Substitute these values back into the determinant expression, simplify the resulting equation, and set it equal to \( -8 \). Then solve the resulting linear equation for \( x \).
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Determinant of a 3x3 Matrix
The determinant is a scalar value that can be computed from a square matrix and provides important properties such as invertibility. For a 3x3 matrix, the determinant is calculated using expansion by minors or the rule of Sarrus, involving sums and products of matrix elements. Understanding how to compute this determinant is essential to solving the given equation.
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Determinants of 2×2 Matrices
Solving Equations Involving Variables in Matrices
When a variable appears inside a matrix, such as 'x' in this problem, the determinant becomes an expression in terms of that variable. Solving the equation involves setting the determinant equal to the given value and solving the resulting algebraic equation for x. This requires combining matrix operations with algebraic manipulation.
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05:28Equations with Two Variables
Properties of Determinants and Their Applications
Determinants have properties like linearity and the effect of row operations, which can simplify calculations. Recognizing these properties helps in efficiently computing the determinant and solving for variables. Additionally, the determinant's value can indicate matrix invertibility, which is often relevant in algebra problems.
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4:36Determinants of 2×2 Matrices