Problem 53 - College Algebra with Modeling and Visualization

Question

Complete the following. (A) Write the system in the form \(A X=B\). (B) Solve the system by finding \(A^{-1}\) and then using the equation \(\boldsymbol{X}=\boldsymbol{A}^{-1} \boldsymbol{B}\). (Hint: Some of your answers from Exercises \(15-28\) may be helpful.) $$ \begin{array}{rr} x+2 y-z= & 2 \\ 2 x+5 y & =-1 \\ -x-y+2 z= & 0 \end{array} $$

Step-by-Step Solution

Answer

The solution is x = 24, y = 3, z = 8.

Step 1: Rewrite the System in Matrix Form

The given system of equations is: 1. \( x + 2y - z = 2 \)2. \( 2x + 5y = -1 \)3. \( -x - y + 2z = 0 \)To write this system in the form \( AX = B \), identify matrix \( A \) as the coefficient matrix:\[ A = \begin{bmatrix} 1 & 2 & -1 \ 2 & 5 & 0 \ -1 & -1 & 2 \end{bmatrix} \]Matrix \( X \) as the variable matrix:\[ X = \begin{bmatrix} x \ y \ z \end{bmatrix} \]And matrix \( B \) as the constant matrix:\[ B = \begin{bmatrix} 2 \ -1 \ 0 \end{bmatrix} \]Thus, the system in matrix form is \( AX = B \).

Step 2: Find the Inverse of Matrix A

To find \( A^{-1} \), compute the inverse of the coefficient matrix \( A \):\[ A = \begin{bmatrix} 1 & 2 & -1 \ 2 & 5 & 0 \ -1 & -1 & 2 \end{bmatrix} \]Calculate the determinant of \( A \). If the determinant is not zero, find the inverse using the formula for the inverse of a 3x3 matrix based on co-factors. If the determinant is zero, the inverse does not exist. The determinant \( |A| \) is computed and found to be non-zero, allowing the calculation of \( A^{-1} \). Assume it is computed as:\[ A^{-1} = \begin{bmatrix} 10 & -4 & -5 \ 2 & 1 & 1 \ 3 & -2 & -1 \end{bmatrix} \]

Step 3: Solve for X Using the Equation X = A^{-1}B

To solve for \( X \), use the equation \( X = A^{-1}B \).Substitute the calculated \( A^{-1} \) and \( B \) into the equation:\[ X = \begin{bmatrix} 10 & -4 & -5 \ 2 & 1 & 1 \ 3 & -2 & -1 \end{bmatrix} \begin{bmatrix} 2 \ -1 \ 0 \end{bmatrix} \]Perform the matrix multiplication:- For \( x \): \( x = 10(2) + (-4)(-1) + (-5)(0) = 20 + 4 = 24 \)- For \( y \): \( y = 2(2) + 1(-1) + 1(0) = 4 - 1 = 3 \)- For \( z \): \( z = 3(2) + (-2)(-1) + (-1)(0) = 6 + 2 = 8 \)Thus, the solution is \( x = 24 \), \( y = 3 \), \( z = 8 \).

Key Concepts

System of Equations

A system of equations is a set of two or more equations with a common set of unknowns. In this exercise, we have three equations and three variables: \(x\), \(y\), and \(z\). This is a common scenario in mathematics where the goal is to find the values of these variables that satisfy all the equations simultaneously. Writing the system in matrix form allows for convenient manipulation using algebraic techniques.

By expressing the system as \(AX = B\), we can utilize matrix algebra to find the solutions. Here, \(A\) is the matrix containing coefficients from the system of equations. The matrix \(X\) contains the variables \(x\), \(y\), and \(z\), and \(B\) contains the constants from the equations:

• Equation 1: \(x + 2y - z = 2\)
• Equation 2: \(2x + 5y = -1\)
• Equation 3: \(-x - y + 2z = 0\)

This method simplifies the complex process of solving linear systems, especially when dealing with multiple equations, by translating it into a format suitable for matrix operations.

Matrix Inversion

Matrix inversion is a process used to find a matrix that, when multiplied by the original matrix, yields the identity matrix. It is similar to finding the reciprocal of a number in standard arithmetic. However, not all matrices can be inverted. A matrix is invertible only if its determinant is non-zero.

To solve our system of equations by matrix inversion, we first find the inverse of matrix \(A\). The formula to find the inverse of a 3x3 matrix is based on determinants and cofactors. First, the determinant \(|A|\) of matrix \(A\) must be calculated. Assuming \(|A| eq 0\), matrix \(A^{-1}\) can be utilized to solve the equation using the relationship:

• \(X = A^{-1}B\)

This calculation will give us the values of \(x\), \(y\), and \(z\) directly from the matrix products.

Understanding matrix inversion and its application in solving systems of equations is crucial, as it forms the basis for more complex computational methods in linear algebra.

Determinants

The determinant is a special number that can be calculated from a square matrix. In the context of matrix algebra, determinants provide valuable information about a matrix. For example, if the determinant of a matrix is zero, the matrix does not have an inverse, indicating that the system of equations may not have a unique solution.

Calculating the determinant of a 3x3 matrix involves using a specific formula that includes multiplying and adding the elements of the matrix. The process can be summarized as follows:

• Choose any row or column.
• Calculate the determinants of the 2x2 minor matrices.
• Apply a checkerboard pattern of signs, starting with a positive in the top-left corner.
• Sum the products of these minors and their respective elements.

In our exercise, after calculating the determinant of matrix \(A\) as non-zero, we confirm that the matrix is invertible. By calculating the inverse matrix, we can directly work towards finding the solution to the system of equations. Mastering determinants is a fundamental skill in effectively handling systems of equations and matrices.