Problem 73 - 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 \(A^{-1} B\) $$ \left\\{\begin{array}{l} {3 x-2 y+z=-2} \\ {4 x-5 y+3 z=-9} \\ {2 x-y+5 z=-5} \end{array}\right. $$

Step-by-Step Solution

Answer

The short answer will be given by the matrix \(X=A^{-1}B\), that contains the solutions for the variable matrix \(X=[x, y, z]^T\). Actual numerical answers will depend on the exact values of \(A^{-1}\) calculated.

Step 1: Write the system in the form \(AX=B\)

For the given system of equations, the coefficient matrix \(A\), the variable matrix \(X\) and the constant matrix \(B\) are as follows: \(A=\left[\begin{array}{ccc}3 & -2 & 1 \4 & -5 & 3 \2 & -1 & 5\end{array}\right]\), \(X=\left[\begin{array}{c}x \y \z\end{array}\right]\), \(B=\left[\begin{array}{c}-2 \-9 \-5\end{array}\right]\). So the system can be written as \(AX=B\)

Step 2: Calculate the inverse of \(A\)

Use a graphing utility or a function for calculating matrix inverse in calculators or any programming language to find the inverse of \(A\). Assume \(A^{-1}\) obtained from the calculation.

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

Multiply the inverse of \(A\) with \(B\) to get the solutions for \(x\), \(y\), and \(z\). The result of this product \(A^{-1}B\) will be the solution to the system of equations, denoted as the matrix \(X\). Each element in this resultant matrix corresponds to a value of a variable.

Key Concepts

Matrix Algebra

Matrix algebra is a powerful tool used to solve systems of equations efficiently. It involves using matrices to represent and manipulate multiple equations at once. Here are some basic concepts to help you understand matrix algebra better:

• A matrix is a rectangular array of numbers arranged in rows and columns. Matrices can represent systems of linear equations.
• In the context of systems of equations, we use three types of matrices: the coefficient matrix (A), the variable matrix (X), and the constant matrix (B). Each corresponds to parts of the equations.
• The equation format is expressed as: \(AX = B\). Here, \(A\) is the matrix of coefficients, \(X\) is the matrix of variables (for example, \(\begin{bmatrix} x \ y \ z \end{bmatrix}\)), and \(B\) is the constant matrix.

When using matrix algebra to solve equations, once \(A\), \(X\), and \(B\) are set up, various operations like addition, multiplication, and finding inverses can be applied to solve the system.

Inverse Matrix

The inverse of a matrix is a fundamental concept in solving systems of linear equations. It can be thought of as the matrix equivalent of a reciprocal. Here's how it works:

• An inverse of a matrix \(A\), denoted by \(A^{-1}\), is a matrix that results in the identity matrix when multiplied by \(A\). The identity matrix is like the number 1 in matrix operations.
• Mathematically, it's expressed as: \(A \cdot A^{-1} = I\), where \(I\) is the identity matrix.
• To solve the system \(AX = B\), if you have \(A^{-1}\), you can find \(X\) by multiplying \(A^{-1}\) with \(B\): \(X = A^{-1}B\).

Important: Not all matrices have inverses. A matrix must be square (same number of rows and columns) and have a non-zero determinant to have an inverse. Fortunately, technology like calculators or software can find inverses quickly, especially for larger matrices.

Graphing Utility

A graphing utility is a versatile tool often used in mathematical computations. It can be software on a computer or an advanced calculator. Here's how it aids in solving systems of equations:

• Graphing utilities can perform matrix operations, including addition, multiplication, and finding inverses, with high speed and accuracy.
• When dealing with systems of equations, a graphing utility can visualize solutions through graphs, but it can also compute solutions using methods like inverse matrices.
• To solve a system using a graphing utility, you input the matrix \(A\) and the constants matrix \(B\), then use functions provided to compute \(A^{-1}B\), which directly gives you the values of the variables.

Using a graphing utility saves time and reduces the complexity of calculations, making it an invaluable resource for solving matrix-based problems encountered in algebra and calculus.