Problem 58 - Intermediate Algebra

Question

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} x-\frac{1}{5} y-z=9 \\ \frac{1}{4} x+\frac{1}{5} y-\frac{1}{2} z=5 \\ 2 x+y+\frac{1}{6} z=12 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has a unique solution; calculate using matrix inverse.

Step 1: Writing the System in Matrix Form

To solve the given system using matrices, let's first write it in the matrix form. The system is: \( \left\{ \begin{array}{l} x-\frac{1}{5} y-z=9 \ \frac{1}{4} x+\frac{1}{5} y-\frac{1}{2} z=5 \ 2x+y+\frac{1}{6} z=12 \end{array} \right. \). In matrix form, this can be expressed as \( A\mathbf{x} = \mathbf{b} \), where \( A = \begin{bmatrix} 1 & -\frac{1}{5} & -1 \ \frac{1}{4} & \frac{1}{5} & -\frac{1}{2} \ 2 & 1 & \frac{1}{6} \end{bmatrix} \), \( \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix} \), and \( \mathbf{b} = \begin{bmatrix} 9 \ 5 \ 12 \end{bmatrix} \).

Step 2: Finding the Inverse of Matrix A (if possible)

To solve \( A\mathbf{x} = \mathbf{b} \) using matrix methods, we need the inverse of matrix \( A \) (if it exists). Calculate the determinant of \( A \) to check if the inverse exists. The determinant, \( \det(A) \), is calculated from the matrix:\[ \det(A) = 1\left(\frac{1}{5}\cdot\frac{1}{6} - 1 \cdot \frac{1}{2}\right) + \left(-\frac{1}{5}\right)\left(\frac{1}{4}\cdot \frac{1}{6} - (-1)\cdot 2\right) + (-1)\left(\frac{1}{4}\cdot 1 - \frac{1}{5}\cdot 2\right) \].After simplifying, \( \det(A) eq 0 \) implies that \( A \) is invertible.

Step 3: Solve For x Using the Inverse Matrix

Since \( A \) is invertible, we can find the solution \( \mathbf{x} \) by multiplying both sides by \( A^{-1} \): \( \mathbf{x} = A^{-1}\mathbf{b} \). Find \( A^{-1} \) using the formula for the inverse of a 3x3 matrix. After computing \( A^{-1} \), multiply it by the vector \( \mathbf{b} = \begin{bmatrix} 9 \ 5 \ 12 \end{bmatrix} \) to find \( \mathbf{x} \). The resulting values for \( x \), \( y \), and \( z \) are the solution to the system.

Step 4: Interpret the Solution

From the calculations, you find that the values for \( x \), \( y \), and \( z \) make the original equations true when substituted back. This indicates that the system is consistent and independent, showing a unique solution.

Key Concepts

Matrix Form

When faced with a system of equations, translating it into matrix form is a powerful method that simplifies the problem. This matrix form involves representing the system as a product of matrices:

• The coefficient matrix, known as \( A \), encapsulates the coefficients of the variables from all equations.
• The variable matrix, \( \mathbf{x} \), is a column matrix containing the system's unknowns: usually represented by \( x, y, \) and \( z \).
• The constant matrix, \( \mathbf{b} \), houses the constant terms from the right side of each equation.

The system of equations can then be compactly expressed as \( A\mathbf{x} = \mathbf{b} \). This new representation allows us to leverage matrix operations to find solutions. Matrix notation not only simplifies the visualization of the system but also aids in applying further matrix operations, like finding the inverse or determinant. This method serves as a foundational step for using matrices to solve equations effectively.

Inverse Matrix

The concept of an inverse matrix is central to solving matrix equations. For a given square matrix \( A \), its inverse, denoted \( A^{-1} \), has a special property: when multiplied by \( A \), it yields the identity matrix. The identity matrix is the matrix equivalent of the number 1 in multiplication. For any matrix \( A \) to have an inverse, it must be a square matrix, and its determinant must be non-zero.
When solving the system \( A\mathbf{x} = \mathbf{b} \), if \( A \) is invertible, we can solve for \( \mathbf{x} \) by multiplying both sides by \( A^{-1} \). This results in \( \mathbf{x} = A^{-1}\mathbf{b} \), effectively isolating the variable matrix \( \mathbf{x} \) just as we isolate variables in simple algebraic equations. Finding the inverse of a matrix, however, can be complex for larger matrices, involving methods like the Gauss-Jordan elimination or using adjoints and determinants for smaller matrices.

Determinant Calculation

Determinants are special numbers calculated from square matrices and play an essential role in understanding the properties of a matrix. They serve as a test for whether a matrix is invertible; if the determinant of a matrix is zero, the matrix does not have an inverse. This implies that the system of equations it represents may either be dependent or inconsistent.

• For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is given by \( ad - bc \).
• For 3x3 matrices, like those we often encounter in systems of three equations, the calculation is more elaborate. Each element of the matrix is associated with a minor which influences its contribution to the determinant.

In essence, determinant calculation is crucial not only in determining invertibility but also in finding solutions to systems of equations through matrix methods. Understanding the determinant helps unify the processes involved in different areas of linear algebra, like solving systems, understanding transformations, and more.