Problem 80 - Precalculus with Limits

Question

WRITING Explain in your own words how to write a system of three linear equations in three variables as a matrix equation, \(AX=B\), as well as how to solve the system using an inverse matrix.

Step-by-Step Solution

Answer

To form the matrix equation, coefficients of the variables form matrix \(A\) and the constants form matrix \(B\). The solution is obtained by multiplying the inverse of \(A\) with \(B\), such that \(X = A^{-1}B\). To find the inverse, use the formula \((1/|A|)adj(A)\).

Step 1: Transforming to Matrix Equation

Start by identifying the coefficients of the variables in each of the three equations. Write these coefficients into a square matrix \(A\), forming the three rows of the matrix. The constants on the right side of each equation are written into a column matrix \(B\). In this way, the system of equations is represented as \(AX = B\). Here, \(X\) is the column matrix containing the variables of the equations.

Step 2: Finding the Inverse Matrix

The next step is to find the inverse of matrix \(A\), denoted as \(A^{-1}\). This is done by using the formula \((1/|A|)adj(A)\), where |A| is the determinant of \(A\) and adj(A) is the adjugate of \(A\).

Step 3: Solving the System

To find \(X\), multiply both sides of the initial matrix equation by the inverse of matrix \(A\). Because an inverse matrix multiplied by its original matrix results in an Identity matrix, this simplifies to \(X = A^{-1}B\). Multiply the inverse matrix with the matrix \(B\) to find the solution matrix \(X\) which contains the solutions to the system of equations.

Key Concepts

Solving Systems of Equations

Solving systems of equations is a fundamental technique in mathematics used for finding values of variables that satisfy multiple equations simultaneously. For instance, a system with three linear equations in three variables can be solved effectively using matrix equations. Understanding this concept involves recognizing a set of linear equations as a unified collection that can be expressed in a compact form. This form is typically a matrix equation, which simplifies the entire problem solving process significantly.
When given a system of linear equations, the first step is to arrange them so that you can identify the coefficients of each variable. These coefficients will form the matrix \(A\). The resulting function looks like \(AX = B\), where \(A\) is the matrix of coefficients, \(X\) is a column matrix or vector of the variables, and \(B\) is a column matrix containing the constants from each equation.

• Identify coefficients from each linear equation.
• Formulate matrix \(A\) using these coefficients.
• Create matrix \(B\) using the constants from the equations.

This matrix equation represents the system and allows us to apply more advanced techniques, such as inverse matrices, to find solutions more efficiently.

Inverse Matrices

The concept of inverse matrices is crucial in solving systems of linear equations through matrix operations. Essentially, an inverse matrix, denoted as \(A^{-1}\), plays a similar role to the reciprocal of a number in arithmetic; it "undoes" the effects of multiplication by the matrix \(A\). To solve \(AX = B\), we need to find \(A^{-1}\).
Finding \(A^{-1}\) involves calculating the determinant \(|A|\) and the adjugate of \(A\), known as \(\text{adj}(A)\). The formula for the inverse matrix is \(A^{-1} = \frac{1}{|A|} \text{adj}(A)\).

• Calculate the determinant \(|A|\).
• Find the adjugate matrix of \(A\).
• Use the formula to find \(A^{-1}\).

Using these steps, once you find \(A^{-1}\), you can solve the system of equations by multiplying both sides of the matrix equation by \(A^{-1}\). This yields the solution matrix \(X\) as \(X = A^{-1}B\). This technique is powerful because if the inverse exists, it provides a unique solution to the system.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, and linear transformations, making it essential for understanding matrix operations and their applications in solving systems of equations. In simple terms, linear algebra provides the tools for dealing with linear systems and is widely applied in various fields such as engineering, physics, and computer science.
In the context of solving systems of equations, linear algebra introduces matrices, which are arrays of numbers that can represent coefficients in linear equations. Matrices can be manipulated similarly to numbers in arithmetic, making operations like addition, subtraction, and multiplication possible. This property is valuable because it allows us to handle and solve large systems of equations efficiently.

• Matrices represent systems of equations succinctly.
• Matrix operations simplify solving complex systems.
• Applications extend beyond pure mathematics.

By understanding linear algebra, learners can grasp how matrices and their properties facilitate solving equations, transforming what may initially seem like an unwieldy problem into a more manageable one through organized structure and predictable operations. This understanding is the cornerstone of modern mathematical problem solving.