Problem 31 - College Algebra

Question

For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$\begin{aligned}-3 x-4 y &=9 \\ 12 x+4 y &=-6 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{3}\) and \(y = -\frac{11}{2}\).

Step 1: Identify the Coefficient Matrix

The given system of equations is: \[-3x - 4y = 9\] \[12x + 4y = -6\] The coefficients of these equations form the matrix \(A\) as follows:\[A = \begin{pmatrix} -3 & -4 \ 12 & 4 \end{pmatrix}\]

Step 2: Find the Inverse of the Coefficient Matrix

To find the inverse of the matrix \(A\), use the formula for a \(2 \times 2\) matrix inverse:\[A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix}\]For \(A = \begin{pmatrix} -3 & -4 \ 12 & 4 \end{pmatrix}\), calculate the determinant \(det(A) = (-3)(4) - (-4)(12) = -12 + 48 = 36\). Thus, \[A^{-1} = \frac{1}{36}\begin{pmatrix} 4 & 4 \ -12 & -3 \end{pmatrix} = \begin{pmatrix} \frac{1}{9} & \frac{1}{9} \ -\frac{1}{3} & -\frac{1}{12} \end{pmatrix}\]

Step 3: Write the Equations in Matrix Form

Now, express the system of equations in matrix form: \[AX = B\] where \[X = \begin{pmatrix} x \ y \end{pmatrix}\] and \[B = \begin{pmatrix} 9 \ -6 \end{pmatrix}\].Thus the matrix equation is: \[\begin{pmatrix} -3 & -4 \ 12 & 4 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 9 \ -6 \end{pmatrix}\].

Step 4: Apply the Inverse

To solve for \(X\), multiply both sides of the equation \(AX = B\) by \(A^{-1}\):\[A^{-1}AX = A^{-1}B\]Since \(A^{-1}A = I\) (where \(I\) is the identity matrix), this simplifies to:\[X = A^{-1}B\]Substitute \(A^{-1}\) and \(B\):\[X = \begin{pmatrix} \frac{1}{9} & \frac{1}{9} \ -\frac{1}{3} & -\frac{1}{12} \end{pmatrix} \begin{pmatrix} 9 \ -6 \end{pmatrix}\]

Step 5: Calculate the Solution

Perform the multiplication:\[X = \begin{pmatrix} \frac{1}{9} \times 9 + \frac{1}{9} \times (-6) \ -\frac{1}{3} \times 9 - \frac{1}{12} \times (-6) \end{pmatrix} = \begin{pmatrix} 1 - \frac{2}{3} \ -3 + \frac{1}{2} \end{pmatrix}\]Simplify each element:\[\begin{pmatrix} \frac{3}{3} - \frac{2}{3} \ -\frac{6}{2} + \frac{1}{2} \end{pmatrix} = \begin{pmatrix} \frac{1}{3} \ -\frac{11}{2} \end{pmatrix}\].Thus, \(x = \frac{1}{3}\) and \(y = \frac{-11}{2}\).

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with matrices, which are essential tools for representing and solving linear equations. Matrices allow you to aggregate and manipulate complex numerical data succinctly. In problems involving matrix algebra, you often perform operations such as addition, subtraction, scalar multiplication, and importantly, finding the inverse of a matrix.
When solving systems of equations, especially linear ones, matrices can simplify the process. In particular, using the inverse of a matrix streamlines finding solutions. The inverse of a matrix, when multiplied with the original matrix, yields the identity matrix - a matrix acting as the number "1" in matrix algebra. The identity matrix maintains the original value of a matrix when multiplied, critical in simplifying and solving matrix equations.
Understanding matrix algebra, therefore, equips you with the necessary tools to tackle complex linear systems efficiently.

2x2 Matrix

A 2x2 matrix is a simple square matrix containing four elements, arranged in two rows and two columns. These matrices are pivotal in linear algebra due to their simplicity yet sufficient complexity to model real-world systems. In terms of computation, 2x2 matrices serve as a building block for larger matrices and more complex operations.
One of the common operations with 2x2 matrices is determining their inverse. The inverse helps solve systems of linear equations by obtaining a resultant matrix with straightforward methods. For a given matrix \[\begin{pmatrix}a & b \c & d \end{pmatrix}\], the inverse exists only if the determinant (ad-bc) is non-zero, implying that the matrix can be reduced to the identity matrix.
The process of working with inverse matrices and determinants requires firm comprehension of multiplication rules and operations unique to matrix algebra.

Systems of Equations

Systems of equations involve finding one or more unknown variables, where each solution needs to satisfy all equations simultaneously. Each linear equation represents a line whose intersection corresponds to the potential solutions for the variables.
Using matrices is one of the efficient methods to simplify and solve systems, especially when dealing with multiple equations and variables. By representing the system in matrix form, you can leverage matrix operations to find solutions, provided that the system is consistent.
In the context of a 2x2 system, the matrix formulation is straightforward. The matrices allow terms to be manipulated collectively with consistent rules, making it easier to solve for unknowns than dealing with individual equations.

Determinants

Determinants are numerical values derived from a square matrix, providing insights into certain properties of the matrix. For a 2x2 matrix, the determinant is calculated as \(ad - bc\). This value is crucial in understanding whether a matrix is invertible.
If the determinant is zero, the matrix does not have an inverse, indicating that the system of equations represented by the matrix doesn't have a unique solution or that the equations are dependent. Conversely, a non-zero determinant implies the matrix can be inverted, thus enabling solutions to a system of equations to be determined.
Determinants have broader applications beyond inverse matrices, such as finding the area of geometric shapes or optimizing certain values in calculus. Their role in multiplying and solving matrix equations renders them an integral concept in matrix algebra.