Textbook Question
Write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.
Ch. 6 - Matrices and Determinants
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 7, Problem 33
In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant.
Verified step by step guidance1
Write down the given 3x3 determinant matrix as:
\[\begin{vmatrix} 1 & 5 & 6 \\ 1 & 4 & 5 \\ 1 & 9 & 10 \end{vmatrix}\]
Use the alternative method for evaluating third-order determinants, which involves expanding along the first row and calculating the minors. The formula is:
\[\text{Determinant} = a(ei - fh) - b(di - fg) + c(dh - eg)\]
where the matrix is:
\[\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\]
Identify the elements from the matrix:
- \(a = 1\), \(b = 5\), \(c = 6\)
- \(d = 1\), \(e = 4\), \(f = 5\)
- \(g = 1\), \(h = 9\), \(i = 10\)
Calculate each of the three minors:
- \(M_1 = ei - fh = (4)(10) - (5)(9)\)
- \(M_2 = di - fg = (1)(10) - (5)(1)\)
- \(M_3 = dh - eg = (1)(9) - (4)(1)\)
Substitute the values back into the determinant formula:
\[\text{Determinant} = a \times M_1 - b \times M_2 + c \times M_3\]
This will give you the value of the determinant after performing the arithmetic.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Third-Order Determinants
A third-order determinant is a scalar value calculated from a 3x3 matrix. It helps determine properties like matrix invertibility and solutions to systems of equations. The determinant is computed using specific methods such as expansion by minors or the alternative method.
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Determinants of 3×3 Matrices
Alternative Method for Evaluating Determinants
The alternative method, often called the diagonal or Sarrus' rule, is a shortcut for calculating 3x3 determinants. It involves summing the products of diagonals from left to right and subtracting the products of diagonals from right to left, simplifying the calculation process.
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Matrix Representation of Systems of Equations
Matrices represent systems of linear equations compactly, where each row corresponds to an equation and each column to coefficients of variables. Understanding this helps relate determinants to the solvability and behavior of the system.
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Related Practice
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Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
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Textbook Question
Write each matrix equation as a system of linear equations without matrices.