Problem 49 - Precalculus: Mathematics for Calculus

Question

Solving a Linear System Solve the system of equations by converting to a matrix equation. Use a graphing calculator to perform the necessary matrix operations, as in Example 7. $$\left\\{\begin{array}{l}12 x+\frac{1}{2} y-7 z=21 \\\11 x-2 y+3 z=43 \\\13 x+y-4 z=29\end{array}\right.$$

Step-by-Step Solution

Answer

Solve the system by calculating the matrix inverse and multiplying by the constants matrix.

Step 1: Write the System as a Matrix Equation

Express the system of equations in matrix form, which is in the form \( AX = B \). Here, \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix:\[ A = \begin{bmatrix} 12 & \frac{1}{2} & -7 \ 11 & -2 & 3 \ 13 & 1 & -4 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} 21 \ 43 \ 29 \end{bmatrix} \]So the matrix equation is:\[ \begin{bmatrix} 12 & \frac{1}{2} & -7 \ 11 & -2 & 3 \ 13 & 1 & -4 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 21 \ 43 \ 29 \end{bmatrix} \]

Step 2: Use the Matrix Inverse to Solve for X

To solve for \( X \), use the formula \( X = A^{-1}B \) if \( A \) is invertible. First, check if \( A \) is invertible by computing its determinant. If the determinant is non-zero, find the inverse of \( A \) using a graphing calculator or a mathematical software. Then compute the product \( A^{-1}B \).

Step 3: Calculate the Determinant of A

Using a graphing calculator, calculate the determinant of matrix \( A \). If the determinant \( \text{det}(A) eq 0 \), \( A \) is invertible, and you can proceed to find \( A^{-1} \). If \( \text{det}(A) = 0 \), the system has no unique solution.

Step 4: Compute A^{-1} and Multiply by B

Assuming \( A \) is invertible (determinant is non-zero), calculate the inverse \( A^{-1} \) using your graphing calculator. Then multiply \( A^{-1} \) by \( B \):\[ X = A^{-1}B \]This step gives you the solution for \( x \), \( y \), and \( z \).

Step 5: Interpret the Results

Finally, interpret the results obtained from the multiplication to find the values of \( x \), \( y \), and \( z \). The resulting matrix will be of the form \( \begin{bmatrix} x \ y \ z \end{bmatrix} \), providing the solutions to the system of equations.

Key Concepts

Matrix Equations

When dealing with systems of linear equations, a convenient method is to express them in the form of a matrix equation. This involves using matrices to represent the coefficients of the variables and the constants from equations. For instance, in our exercise, the equation

• \(12x + \frac{1}{2}y - 7z = 21\)
• \(11x - 2y + 3z = 43\)
• \(13x + y - 4z = 29\)

is transformed into a matrix equation of the form \(AX = B\). Here,

• \(A\) is the coefficient matrix holding the coefficients of the variables \(\begin{bmatrix} 12 & \frac{1}{2} & -7 \ 11 & -2 & 3 \ 13 & 1 & -4 \end{bmatrix}\).
• \(X\) is the variable matrix \(\begin{bmatrix} x \ y \ z \end{bmatrix}\), containing all the unknowns.
• \(B\) is the constant matrix \(\begin{bmatrix} 21 \ 43 \ 29 \end{bmatrix}\) from the equations' right-hand side.

This setup allows us to apply matrix operations to find the solution in a structured manner.

Determinant

The determinant of a matrix is a crucial value used in linear algebra, particularly when dealing with solving systems of equations using matrices. It provides important information about the matrix:

• It indicates whether a matrix is invertible. A non-zero determinant means the matrix can have an inverse, which is essential for finding solutions to matrix equations.
• A zero determinant shows that the matrix is singular, suggesting the system may have no unique solutions or may be dependent.

Calculating the determinant of our coefficient matrix \(A\) helps us understand whether we can proceed to find an inverse. Using a tool like a graphing calculator can simplify the calculation, verifying that \(\text{det}(A) eq 0\), meaning it is possible to solve the system.

Matrix Inverse

The inverse of a matrix, represented as \(A^{-1}\), plays a key role in solving systems of linear equations. The primary function of the inverse is to allow us to solve equations of the form \(AX = B\) by rearranging to \(X = A^{-1}B\). The steps to find \(A^{-1}\) include:

• Ensuring the determinant \(\text{det}(A)\) is not zero, as only non-singular matrices have inverses.
• Using matrix operations or a calculator to compute \(A^{-1}\).

Once the inverse is obtained, multiplying it by matrix \(B\) gives the solutions to the original set of equations. This involves matrix multiplication, another area where graphing calculators can be helpful, ensuring accurate and efficient calculation.

Graphing Calculator

A graphing calculator is a powerful tool often used in solving advanced mathematical problems including matrix operations. When working with systems of equations expressed as matrix equations, graphing calculators simplify the process through:

• Computing the determinant swiftly, which is essential for determining invertibility.
• Finding the inverse of matrices without manual calculations, which can be complex and error-prone.
• Executing matrix multiplication to find solutions, such as in \(X = A^{-1}B\).

With these capabilities, graphing calculators are not only a time-saver but also enhance accuracy, making them indispensable when tackling linear systems through matrices. They also help in visualizing solutions, although this particular task focuses heavily on numerical results.