Problem 41 - College Algebra

Question

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$\begin{aligned} \frac{1}{2} x-\frac{1}{5} y+\frac{1}{5} z &=\frac{31}{100} \\\\-\frac{3}{4} x-\frac{1}{4} y+\frac{1}{2} z &=\frac{7}{40} \\\\-\frac{4}{5} x-\frac{1}{2} y+\frac{3}{2} z &=14 \end{aligned}$$

Step-by-Step Solution

Answer

Find the matrix inverse and multiply with the constants vector to solve for \( x, y, z \).

Step 1: Write the System as a Matrix Equation

The given equations can be rewritten in matrix form as:\[AX = B\]Where, \[A = \begin{bmatrix} \frac{1}{2} & -\frac{1}{5} & \frac{1}{5} \ -\frac{3}{4} & -\frac{1}{4} & \frac{1}{2} \ -\frac{4}{5} & -\frac{1}{2} & \frac{3}{2} \end{bmatrix}, X = \begin{bmatrix} x \ y \ z \end{bmatrix}, B = \begin{bmatrix} \frac{31}{100} \ \frac{7}{40} \ 14 \end{bmatrix}\]

Step 2: Find the Inverse of Matrix A

To solve for the unknowns, we need the inverse of matrix \( A \). Compute the determinant of \( A \) and use it to find the inverse \( A^{-1} \). If \( det(A) eq 0 \), calculate the matrix of minors, cofactor, and then the adjugate, finally dividing by the determinant to get \( A^{-1} \).

Step 3: Calculate \( X \) using the inverse

Once \( A^{-1} \) is found, use it to solve for \( X \) by multiplying both sides of the equation \( AX = B \) with \( A^{-1} \):\[X = A^{-1}B\]Perform the matrix multiplication to determine the values of \( x \), \( y \), and \( z \).

Key Concepts

System of Equations

A system of equations is a set of two or more equations that have common variables. In this case, the challenge is to find the values of the variables \( x \), \( y \), and \( z \) that satisfy all the given equations simultaneously.

In the given exercise, you have three equations that form a system:

• \( \frac{1}{2} x - \frac{1}{5} y + \frac{1}{5} z = \frac{31}{100} \)
• \(-\frac{3}{4} x - \frac{1}{4} y + \frac{1}{2} z = \frac{7}{40} \)
• \(-\frac{4}{5} x - \frac{1}{2} y + \frac{3}{2} z = 14 \)

The goal is to solve this system to find the values of \( x \), \( y \), and \( z \). Using matrices and their inverses is a powerful method to achieve this, especially when dealing with more complex equations and larger systems.

Matrix Determinant

The determinant of a matrix is a special number that helps in finding the inverse of the matrix and in determining whether a matrix is invertible. For a \(3 \times 3\) matrix, the determinant provides crucial information.

When calculating the determinant of a matrix \( A \), if \( \text{det}(A) = 0 \), the matrix has no inverse. For our system, it's essential to find \( \text{det}(A) \) as a first step:

• Arrange the matrix \( A \) as given in the exercise.
• Use the formula for a \(3 \times 3\) matrix determinant which involves summing and subtracting products of the matrix elements.

If the determinant is non-zero, you can proceed to find the inverse, which is needed to solve the system of equations.

Matrix Multiplication

Matrix multiplication is the process of multiplying matrices together. It’s an operation that combines two matrices to produce another matrix. For this exercise, once you find the inverse of \( A \), matrix multiplication helps you solve \( AX = B \).

Here's how you do it:

• Calculate \( A^{-1}B \) by taking each row of \( A^{-1} \) and multiplying it with each column of \( B \).
• Sum the products of corresponding entries from the row and column to get each element of the resulting matrix.

This step ultimately gives you the matrix \( X \), which contains the values for \( x \), \( y \), and \( z \). This shows you not only a solution to the equations but also demonstrates the power of using matrix multiplication in systems of equations.

Step-by-Step Solution

A step-by-step solution helps break down complex procedures into simpler, more manageable parts. Let's apply this concept to solving a system of equations with a \(3 \times 3\) matrix.

For this problem:

• Step 1: Convert the system of equations to a matrix equation \( AX = B \).
• Step 2: Calculate the determinant of \( A \). If it's non-zero, find the inverse \( A^{-1} \) using minors, cofactor, adjugate formulas, and divide by the determinant.
• Step 3: Multiply \( A^{-1} \) by \( B \) to find \( X \), i.e., \( X = A^{-1}B \).

Approaching the solution this way ensures you don’t skip crucial details and can follow along logically step by step, leading you to the correct answer for \( x \), \( y \), and \( z \) in the system.