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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 15
Chapter 7, Problem 15

For Exercises 11–22, use Cramer's Rule to solve each system. {4x−5y=172x+3y=3\(\begin{cases}\)4x - 5y = 17 \\2x + 3y = 3\(\end{cases}\)

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Write the system of equations in matrix form as \(A\mathbf{x} = \mathbf{b}\), where \(A = \begin{bmatrix} 4 & -5 \\ 2 & 3 \end{bmatrix}\), \(\mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}\), and \(\mathbf{b} = \begin{bmatrix} 17 \\ 3 \end{bmatrix}\).
Calculate the determinant of matrix \(A\), denoted as \(\det(A)\), using the formula \(\det(A) = a_{11}a_{22} - a_{12}a_{21}\). For this matrix, compute \(\det(A) = 4 \times 3 - (-5) \times 2\).
Form matrix \(A_x\) by replacing the first column of \(A\) with vector \(\mathbf{b}\), so \(A_x = \begin{bmatrix} 17 & -5 \\ 3 & 3 \end{bmatrix}\). Then calculate \(\det(A_x)\) using the determinant formula.
Form matrix \(A_y\) by replacing the second column of \(A\) with vector \(\mathbf{b}\), so \(A_y = \begin{bmatrix} 4 & 17 \\ 2 & 3 \end{bmatrix}\). Then calculate \(\det(A_y)\) using the determinant formula.
Use Cramer's Rule to find the solutions: \(x = \frac{\det(A_x)}{\det(A)}\) and \(y = \frac{\det(A_y)}{\det(A)}\). These fractions give the values of \(x\) and \(y\) that solve the system.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies when the system has the same number of equations as unknowns and the determinant of the coefficient matrix is non-zero. Each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.
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Determinants of 2x2 Matrices

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This value helps determine if the system has a unique solution (non-zero determinant) or not. Determinants are essential in Cramer's Rule to find the values of variables by comparing determinants of modified matrices.
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Solving Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, and matrix approaches like Cramer's Rule, which is efficient for small systems.
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Related Practice
Textbook Question

In Exercises 9 - 16, find the following matrices: c. - 4A

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Textbook Question

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

Textbook Question

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B

Textbook Question

Use the fact that if A=[abcd]A = \(\begin{bmatrix}\) a & b \\ c & d \(\end{bmatrix}\), then A−1=1ad−bc[d−b−ca]A^{-1} = \(\frac{1}{ad-bc}\) \(\begin{bmatrix}\) d & -b \\ -c & a \(\end{bmatrix}\) to find the inverse of each matrix, if possible. Check that AA−1=I2AA^{-1} = I_2 and A−1A=I2A^{-1} A = I_2.

A=[3−1−42]A = \(\begin{bmatrix}\)3 & -1 \\-4 & 2\(\end{bmatrix}\)

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Textbook Question

Perform each matrix row operation and write the new matrix.

[1−32∣031−1∣72−21∣3]−3R1+R2\(\begin{bmatrix}\)1 & -3 & 2 & \(\vert\) & 0 \\3 & 1 & -1 & \(\vert\) & 7 \\2 & -2 & 1 & \(\vert\) & 3\(\end{bmatrix}\]\quad\) -3R_1 + R_2

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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. {2x+y−z=23x+3y−2z=3\(\begin{cases}\)2x + y - z = 2 \\3x + 3y - 2z = 3\(\end{cases}\)

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