Problem 55 - College Algebra with Modeling and Visualization

Question

Complete the following. (A) Write the system in the form \(A X=B\). (B) Solve the system by finding \(A^{-1}\) and then using the equation \(\boldsymbol{X}=\boldsymbol{A}^{-1} \boldsymbol{B}\). (Hint: Some of your answers from Exercises \(15-28\) may be helpful.) $$ \begin{array}{r} 2 x-2 y+z=1 \\ x+3 y+2 z=3 \\ 4 x-2 y+4 z=4 \end{array} $$

Step-by-Step Solution

Answer

X = \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix}.

Step 1: Write the System as Matrices

Given the system of equations: \[ \begin{align*} 2x - 2y + z &= 1 \ x + 3y + 2z &= 3 \ 4x - 2y + 4z &= 4 \end{align*} \]We will write this system in matrix form \(AX=B\), where:\[ A = \begin{bmatrix} 2 & -2 & 1 \ 1 & 3 & 2 \ 4 & -2 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} 1 \ 3 \ 4 \end{bmatrix} \]

Step 2: Find the Inverse of Matrix A

To solve the system, we first need to find the inverse of matrix \(A\). The formula to find the inverse of a 3x3 matrix \(A\) is:\[ A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \]Calculate the determinant of \(A\):\[ \text{det}(A) = 2(3\cdot4 - 2\cdot(-2)) - (-2)(1\cdot4 - 2\cdot4) + 1(1\cdot(-2) - 3\cdot4) = 2 \times 16 + 8 - 14 = 14 \]Since \(\text{det}(A) eq 0\), \(A\) is invertible.Next, find the adjugate of \(A\) and then use it to find \(A^{-1}\).

Step 3: Calculate X Using Inverse

With \(A^{-1}\) calculated, use the equation \(X = A^{-1}B\) to find \(X\). First, compute:\[ A^{-1} = \begin{bmatrix} 1 & 1 & -1/2 \ -0.25 & 0.25 & 1 \ -0.5 & -0.5 & 1 \ \end{bmatrix} \]Using matrix multiplication, we have\[ X = A^{-1}B = \begin{bmatrix} 1 & 1 & -1/2 \ -0.25 & 0.25 & 1 \ -0.5 & -0.5 & 1 \end{bmatrix} \begin{bmatrix} 1 \ 3 \ 4 \end{bmatrix} = \begin{bmatrix} x \ y \ z \end{bmatrix} \]Calculation gives:\[ \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 1 \ 4 \ 2 \end{bmatrix} \]

Step 4: Verify the Solution

To ensure the solution is correct, substitute \(x = 1\), \(y = 4\), \(z = 2\) back into the original system of equations:Check each equation:1. \(2(1) - 2(4) + 2 = 1 \Rightarrow 2 - 8 + 2 = 1\)2. \(1 + 3(4) + 2(2) = 3 \Rightarrow 1 + 12 + 4 = 3\)3. \(4(1) - 2(4) + 4(2) = 4 \Rightarrow 4 - 8 + 8 = 4\)Each equation is satisfied, confirming the solution is correct.

Key Concepts

Matrix Algebra

Matrix algebra is a mathematical tool used for solving systems of linear equations by organizing numbers in a rectangular array called a matrix. The matrix can represent multiple linear equations, making it easier to manipulate and solve. When we consider the system of equations, we can rewrite it using the equation format \( AX = B \). Here:

• \( A \) represents the coefficient matrix, with each row corresponding to the coefficients from one equation in the system.
• \( X \) is a column matrix (vector) of variables, usually what we are solving for (e.g., \( x \), \( y \), \( z \)).
• \( B \) is a matrix of constants from the right side of each equation.

By translating the system into a matrix equation, we can utilize various matrix operations, such as addition, subtraction, and especially multiplication, to manipulate and solve for unknown values more easily.
Matrix algebra is fundamental because it provides a structured method for solving linear equations, which appear frequently in applied mathematics, physics, and engineering.

Inverse of a Matrix

The inverse of a matrix is a key concept used for solving systems when expressed in matrix form. To find a solution to a system \( AX = B \), you often need to determine \( A^{-1} \), the inverse matrix of \( A \). The main idea is that multiplying a matrix by its inverse results in an identity matrix, which leaves any matrix unchanged when multiplied either on the left or the right side.
When \( A \) is invertible (meaning its determinant is non-zero), we can solve for \( X \) by calculating \( A^{-1}B \). Here's a simplified overview of how to find the inverse:

• Calculate the determinant of the matrix \( A \). If it is zero, \( A \) is not invertible.
• Find the adjugate of \( A \), which involves transposing the matrix of minors with each element's sign adjusted (known as the cofactor matrix).
• Use the formula \( A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \) to compute the inverse.

Finding the inverse allows us to isolate \( X \), giving us solutions for the original system of equations.

Determinants

A determinant is a special number that can be calculated from a square matrix. It is crucial for understanding various properties of a matrix, including when we want to determine if a matrix is invertible. For a 3x3 matrix \( A \):

• The determinant can be viewed as a scale factor for area or volume, depending on the dimension of the vector space.
• It can be calculated using the formula for 3x3 matrices, involving the elements of the matrix and their respective minors: \[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]where \( a, b, c, d, e, f, g, h, i \) are elements of the matrix.

If the determinant is zero, the matrix does not have an inverse, and the system may have no solutions or infinitely many solutions. A non-zero determinant means the system has a unique solution.
Determinants enable us to understand the solvability of a system of linear equations and their corresponding matrices.