Problem 24 - Algebra for College Students

Question

Use a matrix approach to solve each system. \(\left(\begin{array}{rl}x-4 y+3 z & =16 \\ 2 x+3 y-4 z & =-22 \\ -3 x+11 y-z & =-36\end{array}\right)\)

Step-by-Step Solution

Answer

Solve using matrix inversion: Find \(A^{-1}B\) to get the values of \(x\), \(y\), and \(z\).

Step 1: Write the System in Matrix Form

The given system can be expressed as a matrix equation. The system is: 1. \(x - 4y + 3z = 16\)2. \(2x + 3y - 4z = -22\)3. \(-3x + 11y - z = -36\). Write this in the form \(AX = B\) where \( A = \begin{pmatrix} 1 & -4 & 3 \ 2 & 3 & -4 \ -3 & 11 & -1 \end{pmatrix}\), \(X = \begin{pmatrix} x \ y \ z \end{pmatrix}\), and \(B = \begin{pmatrix} 16 \ -22 \ -36 \end{pmatrix}\).

Step 2: Calculate the Inverse of Matrix A

To find the solution, compute the inverse of matrix \(A\). Use the formula: \( A^{-1} = \frac{1}{ ext{det}(A)} ext{ adj}(A) \).Calculate the determinant of \(A\) and then the adjugate. Verify that the determinant is non-zero to ensure that \(A\) is invertible. Compute \(A^{-1}\).

Step 3: Multiply the Inverse by B

Calculate \(X\) by multiplying \(A^{-1}\) with \(B\). This gives: \(X = A^{-1}B\). Perform the matrix multiplication to find the values of \(x\), \(y\), and \(z\).

Step 4: Verify the Solution

Substitute \(x\), \(y\), and \(z\) back into the original equations to ensure they satisfy all three equations. This confirms the validity of the solution.

Key Concepts

Inverse Matrix

To solve a system of equations using matrices, one useful tool is the inverse of a matrix. The inverse of a matrix, denoted as \(A^{-1}\), is analogous to a reciprocal in arithmetic. When you multiply a matrix by its inverse, you obtain the identity matrix.
Imagine matrix \(A\) is a set of transformations that you want to reverse to find the original variables \(X\) in the system \(AX = B\). If \(A\) is invertible (meaning it has an inverse), you can multiply both sides by \(A^{-1}\) to solve for \(X\):

• \(AX = B \) becomes \(A^{-1}AX = A^{-1}B \)
• \(IX = A^{-1}B \)
• \(X = A^{-1}B \)

This resulting equation leads you directly to the solution of the system. However, the inverse can only be found if the determinant of \(A\) is not zero, indicating that \(A\) is invertible.

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix and plays a crucial role in understanding the properties of the matrix, most notably its invertibility. For a 3x3 matrix \(A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}\), the determinant is calculated using:
\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This formula helps to determine whether the matrix has an inverse:

• If \(\text{det}(A) eq 0\), the matrix \(A\) is invertible.
• If \(\text{det}(A) = 0\), the matrix \(A\) is singular and does not have an inverse.

When solving a system of equations, if the matrix representing the system has a zero determinant, it means that there is either no solution or infinitely many solutions. Thus, properly calculating the determinant is critical in matrix solution methods.

Adjugate Matrix

The adjugate matrix, sometimes called the adjoint, is used as part of finding the inverse of a matrix. It is the transpose of the cofactor matrix of \(A\). For a 3x3 matrix, the cofactor of an element is computed by taking the determinant of the 2x2 matrix that remains after excluding the row and column of the element.
Here's a simple way to understand it:

• The adjugate matrix is computed by placing these cofactors in a matrix and then taking the transpose.
• The formula for finding the inverse using the adjugate is: \[A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)\]

Remember, calculation of the adjugate requires the determinant of each minor (2x2 submatrix) and then organizing these into a transposed matrix. The adjugate matrix becomes crucial if the determinant is non-zero, allowing you to compute the inverse. Once the inverse is calculated, you can use it to solve the system \(AX = B\) as explained above.