Problem 32 - Algebra for College Students

Question

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{rl} x+9 y & =-5 \\ 4 x-7 y & =-20 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \( x = -3 \) and \( y = -2 \).

Step 1: Write the System in Matrix Form

Express the given system of equations as \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the variable matrix, and \( \mathbf{b} \) is the constants matrix. Here, \( A = \begin{pmatrix} 1 & 9 \ 4 & -7 \end{pmatrix} \), \( \mathbf{x} = \begin{pmatrix} x \ y \end{pmatrix} \), and \( \mathbf{b} = \begin{pmatrix} -5 \ -20 \end{pmatrix} \).

Step 2: Find the Inverse of Matrix A

Calculate the inverse of matrix \( A = \begin{pmatrix} 1 & 9 \ 4 & -7 \end{pmatrix} \). The inverse, \( A^{-1} \), can be computed using \( A^{-1} = \frac{1}{ ext{det}(A)} \text{adj}(A) \). First, find the determinant: \( \text{det}(A) = (1)(-7) - (9)(4) = -7 - 36 = -43 \). Then, calculate \( A^{-1} = \frac{1}{-43} \begin{pmatrix} -7 & -9 \ -4 & 1 \end{pmatrix} = \begin{pmatrix} \frac{7}{43} & \frac{9}{43} \ \frac{4}{43} & -\frac{1}{43} \end{pmatrix} \).

Step 3: Multiply A^{-1} by b

Multiply \( A^{-1} \) by \( \mathbf{b} \) to find \( \mathbf{x} \). Compute \( \mathbf{x} = A^{-1} \mathbf{b} = \begin{pmatrix} \frac{7}{43} & \frac{9}{43} \ \frac{4}{43} & -\frac{1}{43} \end{pmatrix} \begin{pmatrix} -5 \ -20 \end{pmatrix} = \begin{pmatrix} \frac{7}{43}(-5) + \frac{9}{43}(-20) \ \frac{4}{43}(-5) + \left(-\frac{1}{43}\right)(-20) \end{pmatrix} = \begin{pmatrix} -3 \ \ -2 \end{pmatrix} \).

Step 4: Interpret the Solution

The solution \( \mathbf{x} = \begin{pmatrix} -3 \ -2 \end{pmatrix} \) indicates that \( x = -3 \) and \( y = -2 \). This is the final solution to the system of equations.

Key Concepts

Systems of Equations

A system of equations consists of multiple equations that share common variables. In solving systems of equations, we aim to find values for the variables that satisfy all the given equations simultaneously. Systems can be either linear or nonlinear, depending on the nature of the equations involved.

In the context of matrix inverses, linear equations are particularly significant because they can be expressed in matrix form. A typical system of linear equations can be represented as a matrix equation \(A\mathbf{x} = \mathbf{b}\), where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the variable matrix, and \(\mathbf{b}\) is the constants matrix. Solving the system means finding the variable matrix \(\mathbf{x}\), which contains the values of unknowns.

Solving such systems using matrix inverses is efficient, especially when dealing with larger systems. The matrix inverse method involves finding the inverse of the coefficient matrix and using it to isolate \(\mathbf{x}\). This approach can provide an exact solution if the coefficient matrix is invertible.

Coefficient Matrix

The coefficient matrix is a crucial part of solving linear systems using matrices. It is constructed by taking the coefficients of the variables from each equation in the system and arranging them into a square matrix. For example, in the equation system:

• \(x + 9y = -5\)
• \(4x - 7y = -20\)

The coefficient matrix \(A\) is:

• \[ A = \begin{pmatrix} 1 & 9 \ 4 & -7 \end{pmatrix} \]

The coefficient matrix plays a fundamental role in determining the solvability of the system. It is used to form the equation \(A\mathbf{x} = \mathbf{b}\). Understanding the properties of the coefficient matrix, such as invertibility, is vital because an inverse can't be found if the matrix is non-invertible.

If the determinant of the coefficient matrix is non-zero, the matrix is invertible, implying that a unique solution exists. Otherwise, the system may have no solution or infinitely many solutions.

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides essential insights into the properties of the matrix and the associated system of equations. For a 2x2 matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is computed as:

• \( \text{det}(A) = ad - bc \)

In our example, the coefficient matrix

• \( A = \begin{pmatrix} 1 & 9 \ 4 & -7 \end{pmatrix} \)

Has a determinant of \( -43 \), given by:

• \( \text{det}(A) = (1)(-7) - (9)(4) = -7 - 36 = -43 \)

The significance of the determinant extends to determining the existence and uniqueness of solutions for the system. A non-zero determinant indicates that the matrix is invertible, making it possible to solve the system uniquely using matrix inversion. Conversely, a determinant of zero suggests that the matrix is singular, and solutions might not be possible, or they aren't unique.