Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Use an inverse matrix to solve (if possible) the system of linear equations. $$ \left\\{\begin{array}{rr} 4 x-2 y+3 z= & -2 \\ 2 x+2 y+5 z= & 16 \\ 8 x-5 y-2 z= & 4 \end{array}\right. $$

Step-by-Step Solution

Answer

The solutions to the system of equations are the elements in the solution matrix \( X \) after multiplying the inverse of matrix \( A \), \( A^{-1} \), with the constant matrix \( B \).

Step 1: Formulate The Matrix System

Rearrange the given system of equations as a matrix equation, i.e., \( AX = B \). The coefficient matrix \( A \), the variable matrix \( X \) and the constant matrix \( B \) will be as follows: \[ A = \left[ \begin{array}{ccc} 4 & -2 & 3 \ 2 & 2 & 5 \ 8 & -5 & -2 \end{array} \right] \] \[ X = \left[ \begin{array}{c} x \ y \ z \end{array} \right], B = \left[ \begin{array}{c} -2 \ 16 \ 4 \end{array} \right] \].

Step 2: Check the Invertibility of Matrix A

Calculate the determinant of the coefficient matrix \( A \). If the determinant is not zero, matrix \( A \) is invertible, which means our system of equations has a unique solution.

Step 3: Calculate the inverse of A

If the matrix is invertible, calculate the inverse of the coefficient matrix \( A \), denoted by \( A^{-1} \). This can be done using Gaussian elimination method, Using adjoint method or using minor and cofactor method.

Step 4: Solve for X

Multiply the inverse of matrix \( A \), \( A^{-1} \), with the constant matrix \( B \) to get the solution matrix \( X \), i.e.,\( X = A^{-1}B \). The elements of solution matrix \( X \) are the solutions for variables \( x, y, z \), respectively.

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple equations that are linear in nature with common variables. In our problem, we need to solve three equations with three unknowns: \( x \), \( y \), and \( z \). Every equation in the system gives a linear relationship among these variables, which can be expressed as:

• \( 4x - 2y + 3z = -2 \)
• \( 2x + 2y + 5z = 16 \)
• \( 8x - 5y - 2z = 4 \)

These equations can be solved simultaneously, and a valid solution will satisfy all equations at the same time. First, they are arranged into a matrix form, typically denoted by \( AX = B \), making it easier to manipulate the equations using matrix operations.

Matrix Inversion

Matrix inversion is a key step in solving systems of equations aligned in matrix form. For the equation \( AX = B \), if matrix \( A \) is invertible, we can find its inverse \( A^{-1} \). Once the inverse is obtained, it can be multiplied with matrix \( B \) to solve for matrix \( X \) which contains our variables:

• \( A^{-1}B = X \)

The condition for matrix inversion is that the matrix must be square (same number of rows and columns) and its determinant should not be zero. If the determinant of \( A \) is zero, it means \( A \) doesn't have an inverse, indicating that the system could have no solution or infinitely many solutions.

Determinant

The determinant provides crucial information about a square matrix. For matrix \( A \), the determinant describes properties such as invertibility. A nonzero determinant means the matrix is invertible, suggesting a unique solution exists for the system of equations.For a \( 3 \times 3 \) matrix, the determinant is calculated through expansion by minors or using the rule of Sarrus. It's essential to compute this value before attempting to find the inverse of a matrix. If we encounter a zero, the matrix cannot be inverted, and we must consider alternative methods, such as row reduction, for analysing the system.

Gaussian Elimination

Gaussian elimination is a systematic method to reduce a matrix to its row-echelon form. This technique helps in determining whether a matrix is invertible and can directly solve systems of linear equations. The process involves:

• Performing row operations to achieve an upper triangular matrix
• Using back-substitution to find the solutions for the variables

When finding an inverse, Gaussian elimination helps transform the matrix into an identity matrix, the inverse of itself, through further row operations. This is particularly useful for larger systems or when finding an exact numerical solution is required. It organizes the steps and simplifies computations while adhering to mathematical rigor.