Problem 76 - Algebra and Trigonometry

Question

Write each system in the form \(A X=B .\) Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\) $$ \left\\{\begin{array}{r} {w+x+y+z=4} \\ {w+3 x-2 y+2 z=7} \\ {2 w+2 x+y+z=3} \\ {w-x+2 y+3 z=5} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution of the given system of equations is \(w=1, x=1, y=1, z=1\).

Step 1: Translate the System of Equations into Matrix Form \(AX=B\)

Each of the coefficients in front of the variables \(w, x, y, z\) fill up a row in the \(A\) matrix and each of the numbers on the right side of the equation fill up a row in the \(B\) matrix. The matrix will be:\(A=\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 3 & -2 & 2 \\ 2 & 2 & 1 & 1 \\ 1 & -1 & 2 & 3 \end{bmatrix}\), \(X=\begin{bmatrix} w \\ x \\ y \\ z \end{bmatrix}\), and \(B=\begin{bmatrix} 4 \\ 7 \\ 3 \\ 5 \end{bmatrix}\)

Step 2: Find the inverse of matrix \(A\)

To find \(A^{-1}\), the inverse of matrix \(A\), use a graphing utility or calculator as the exercise instructed. Upon doing so, we find \(A^{-1}=\begin{bmatrix} 0.6 & 0.2 & -0.4 & 0.2 \\ -1.5 & 0.5 & 1 & -0.5 \\ -1 & 0 & 1 & 0 \\ 2.9 & -0.3 & -2.6 & 0.1 \end{bmatrix}\)

Step 3: Compute \(A^{-1}B\)

Now, multiply the inverse of matrix \(A\) with matrix \(B\). This gives us the solution, which will be a column matrix holding the solutions for \(w, x, y, z\): \(A^{-1}B=\begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}\)

Key Concepts

Matrix Form

When working with a system of equations, it's often helpful to express it in matrix form. This involves creating a matrix representation that succinctly captures all the information of the equations involved. In this format, you typically have three matrices:

• Matrix \(A\), which contains the coefficients of each variable from all equations.
• Matrix \(X\), which stacks all the variables (\(w, x, y, z\) in our example) into a column.
• Matrix \(B\), which houses the constants from the right-hand side of each equation.

For instance, if you have the system of equations given, you can form matrices like this:- \(A = \begin{bmatrix} 1 & 1 & 1 & 1 \ 1 & 3 & -2 & 2 \ 2 & 2 & 1 & 1 \ 1 & -1 & 2 & 3 \end{bmatrix}\)- \(X = \begin{bmatrix} w \ x \ y \ z \end{bmatrix}\)- \(B = \begin{bmatrix} 4 \ 7 \ 3 \ 5 \end{bmatrix}\)
This makes solving the system using matrix operations easier and more systematic.

Inverse Matrix

The inverse matrix is a crucial concept when solving linear equations using matrix methods. An inverse of a matrix \(A\), denoted as \(A^{-1}\), is like a reciprocal in numbers. When you multiply \(A\) by \(A^{-1}\), you end up with the identity matrix, which acts like the number 1 in matrix operations:

• The identity matrix for a 4x4 matrix looks like: \(I = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}\)
• Not every matrix has an inverse. A matrix must be square and of full rank to have an inverse.

Once you have the inverse, you can solve the equation \(AX = B\) by multiplying both sides by \(A^{-1}\) to isolate \(X\):- \(A^{-1}AX = A^{-1}B\)- Since \(A^{-1}A = I\), this simplifies to \(IX = A^{-1}B\) or just \(X = A^{-1}B\).
This gives you the solutions to the variables in matrix \(X\). Calculators and graphing utilities often have functions to find inverses swiftly.

System of Equations

A system of equations is a set of two or more equations with multiple variables. The goal when solving a system is to find values for the variables that satisfy all equations simultaneously. The system laid out in the original exercise is:- \(w + x + y + z = 4\)- \(w + 3x - 2y + 2z = 7\)- \(2w + 2x + y + z = 3\)- \(w - x + 2y + 3z = 5\)
There are multiple methods to solve systems of equations:

• Substitution and elimination are traditional methods for smaller systems.
• Matrix methods like using inverse matrices become handy for larger systems, allowing computational tools to do the heavy lifting.

Understanding how to express systems in matrix form and solve them using matrix operations broadens your toolkit for tackling these problems efficiently.

Graphing Utility

A graphing utility is a versatile tool capable of simplifying complex mathematical procedures, especially in algebra and calculus. When dealing with matrices, a graphing utility can:

• Quickly find the inverse of a matrix, which is a computation that can become tedious by hand for larger matrices.
• Facilitate the multiplication of matrices, which helps when solving \(A^{-1}B\) to find the solution to a system of equations.
• Visualize functions and datasets that aid in comprehending solutions graphically if needed.

While not every graphing utility may support all features, those equipped with matrix operations can significantly reduce the time and effort required to solve a system of equations. This makes learning how to efficiently use such tools invaluable for students tackling advanced mathematics.