Problem 32 - Precalculus: Mathematical for Calculus

Question

Use a calculator that can perform matrix operations to solve the system, as in Example 7. $$\left\\{\begin{array}{l} 3 x+4 y-z=2 \\ 2 x-3 y+z=-5 \\ 5 x-2 y+2 z=-3 \end{array}\right.$$

Step-by-Step Solution

Answer

Use a calculator to calculate \( A^{-1} \) and solve \( X = A^{-1}B \).

Step 1: Write the System as a Matrix Equation

The system of linear equations can be represented in matrix form as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the column matrix of variables, and \( B \) is the column matrix of constants. Thus,\[A = \begin{pmatrix} 3 & 4 & -1 \ 2 & -3 & 1 \ 5 & -2 & 2 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \ z \end{pmatrix}, \quad B = \begin{pmatrix} 2 \ -5 \ -3 \end{pmatrix}\]then \( AX = B \).

Step 2: Use the Calculator to Find the Inverse of A

Calculate the inverse of the coefficient matrix \( A \). Using a calculator, you find that \( A^{-1} \) is \[ A^{-1} = \begin{pmatrix} ice-cream \ shop \ open \end{pmatrix} \]

Step 3: Compute the Solution using Matrix Multiplication

Multiply the inverse of \( A \), \( A^{-1} \), with the constant matrix \( B \) to find \( X \). Use the calculator to perform the multiplication: \( X = A^{-1}B \). The product gives the values for \( x, y, \) and \( z \).

Step 4: Verify the Solution

Substitute the values obtained for \( x, y, \) and \( z \) back into the original equations to ensure they satisfy all three equations. If they do, then the solution is verified as correct.

Key Concepts

Inverse Matrix

An inverse matrix is a mathematical concept used in various calculations, including solving systems of linear equations. Imagine an inverse matrix as a magic key that unlocks the mystery of these systems by reversing their "action."

Only square matrices (same number of rows and columns) can potentially have inverses, and not all square matrices are invertible. If a matrix \( A \) is invertible, there exists another matrix, \( A^{-1} \), such that when you multiply \( A \) by \( A^{-1} \), the result is the identity matrix. The identity matrix behaves much like the number 1 in multiplication.

• Identity Matrix: \( I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} \)
• Matrix Inversion: Typically found using calculators or computer software as it can be a complex and computation-heavy task.

The inverse matrix is used directly in solving the matrix equation \( AX = B \) by rearranging it to \( X = A^{-1}B \). This step is crucial as it allows us to isolate \( X \), the matrix of variables, and solve for specific variable values.

Matrix Multiplication

Matrix multiplication is an essential skill when working with matrices, and it follows its own special set of rules. This process combines rows from the first matrix with columns of the second to create a new matrix. It's very different from multiplying regular numbers, so it's important to keep the steps clear in mind.

• Dimensions Matter! For multiplication, an \( m \times n \) matrix can only be multiplied with an \( n \times p \) matrix.
• Resulting Matrix: If the above condition is met, the resulting matrix will be \( m \times p \).

In our solution: we have multiplied the inverse matrix \( A^{-1} \) with the column matrix \( B \). Each element in the resulting matrix corresponds to a solution for one of the variables in the system of equations. Calculators often have built-in functions for matrix multiplication to simplify this task.

System of Linear Equations

A system of linear equations is a set of equations with the same variables. Each equation forms a straight line, and the solution to the system is the point or points where these lines intersect.

For example, a simple system might be:

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

Here, there are three equations with three variables: \( x, y, \) and \( z \). The goal is to find the values of \( x, y, \) and \( z \) that satisfy all equations simultaneously. Converting these into a matrix form, as shown in the solution, simplifies handling these equations in a more structured, understandable way.

Solving systems of linear equations using matrices, particularly with the inverse matrix method when applicable, offers a robust, mathematically elegant approach to finding solutions, especially for larger systems.