Problem 73 - College Algebra

Question

Write each system in the form \(A X=B\). Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\). $$\left\\{\begin{array}{l}3 x-2 y+z=-2 \\\4 x-5 y+3 z=-9 \\\2 x-y+5 z=-5\end{array}\right.$$

Step-by-Step Solution

Answer

Using the matrix form \(AX = B\) and computing \(A^{-1}B\), We'll have the solution to the system in terms of matrix \(X\). The final solution is dependent on the values calculated by the graphing utility for \(A^{-1}B\).

Step 1: Writing the System in the Form of \(AX = B\)

The given system of equations is: \[\begin{align*} 3x - 2y + z &= -2, \\ 4x - 5y + 3z &= -9, \\ 2x - y + 5z &= -5. \end{align*}\] The coefficients of \(x,y,z\) form the matrix \(A\), and the constants on the right sides of the equations form the column matrix \(B\). Hence, the matrix form will be: \[A = \left[ \begin{array}{ccc} 3 & -2 & 1 \\ 4 & -5 & 3 \\ 2 & -1 & 5 \end{array} \right], X = \left[ \begin{array}{c} x \\ y \\ z \end{array} \right], B = \left[ \begin{array}{c} -2 \\ -9 \\ -5 \end{array} \right].\]

Step 2: Calculating the Inverse of Matrix \(A\)

To solve for \(X\), we need to compute \(A^{-1}\), the inverse of \(A\), and then multiply \(A^{-1}\) and \(B\). This calculation can be executed using a graphing utility.

Step 3: Computing the Solution by Multiplying \(A^{-1}\) and \(B\)

After obtaining \(A^{-1}\), we can multiply it with \(B\). This multiplication will give us the solutions for \(x\), \(y\) and \(z\).

Step 4: Solving the System

Let's suppose the result from \(A^{-1}B\) was \[ X = \left[ \begin{array}{c} x \\ y \\ z \end{array} \right].\] This will denote \(X\) as the solution to the system of equations, where each element of the vector \(X\) represents the solution for each unknown in the original system.

Key Concepts

Matrix Inverse

A matrix inverse is a key player in solving systems of linear equations using algebraic methods. When we have a square matrix, say \( A \), its inverse, denoted as \( A^{-1} \), plays a role similar to division in arithmetic. In simpler terms, if you multiply a matrix by its inverse, you get the identity matrix, similar to how multiplying a number by its reciprocal yields 1.

To find the solution for a system of equations expressed in matrix format \( AX = B \), calculating \( A^{-1} \) allows us to solve for \( X \) using the formula \( X = A^{-1}B \). However, it's crucial to remember that not all matrices have inverses. To check if the inverse exists, the determinant of the matrix must not be zero. If it is zero, the matrix is termed "singular," and another method is needed to find the solution.

Graphing calculators or graphing utilities often have functions that can swiftly compute a matrix inverse, making them very handy when working with multiple systems of equations. These tools eliminate the complexities of manual calculations, enhancing both speed and accuracy.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. Solving such systems means finding the values of these unknowns that satisfy all the equations simultaneously. Linear systems are the most common, where each equation is a straight line if graphed on a plane.

Given the system provided:

• \( 3x - 2y + z = -2 \)
• \( 4x - 5y + 3z = -9 \)
• \( 2x - y + 5z = -5 \)

We arranged our variables and constants into matrices. Here, matrix \( A \) contains the coefficients of \( x, y, \) and \( z \), matrix \( X \) embodies the unknowns \( x, y, \) and \( z \), and matrix \( B \) includes constants from the equations' right-hand sides.

Transforming the system of equations into matrix form makes it much easier to leverage algebraic techniques and computational tools, such as graphing utilities, to find solutions. By solving the system, we are essentially determining the point at which all the planes described by each equation meet.

Graphing Utility

Graphing utilities are fantastic tools for solving complex mathematical problems like systems of equations. These tools can graph functions, solve equations, and, crucially, perform matrix operations like finding the inverse of matrices and multiplying matrices.

To solve the system of linear equations from our example using a graphing utility, we follow these steps:

• Enter matrix \( A \) and matrix \( B \) into the utility.
• Use the utility's function to calculate \( A^{-1} \).
• Multiply \( A^{-1} \) by \( B \) to gain the solution matrix \( X \).

Graphing utilities perform these operations quickly and accurately, making them an invaluable resource for checking manual solutions or handling cumbersome calculations. By automating the matrix manipulations, graphing utilities provide clear insights and visualizations, perfect for both verifying theoretical solutions and exploring practical applications.