Problem 30 - Algebra for College Students

Question

Use a matrix approach to solve each system. \(\left(\begin{array}{r}4 x+3 y-z=0 \\ 3 x+2 y+5 z=6 \\ 5 x-y-3 z=3\end{array}\right)\)

Step-by-Step Solution

Answer

Use matrix inversion to solve the system and verify by substitution.

Step 1: Write System as a Matrix Equation

Start by expressing the system of equations in matrix form, which is done as follows: \( A \cdot \mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the vector of constants. For our system:\[ A = \begin{bmatrix} 4 & 3 & -1 \ 3 & 2 & 5 \ 5 & -1 & -3 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 0 \ 6 \ 3 \end{bmatrix} \] Thus, the matrix equation is: \( A \cdot \mathbf{x} = \mathbf{b} \).

Step 2: Calculate Inverse of Coefficient Matrix

The solution vector \( \mathbf{x} \) can be found using the inverse of matrix \( A \) if \( A \) is invertible. First, find \( A^{-1} \), the inverse of \( A \). Compute the determinant of \( A \); ensure it is non-zero to continue. If det(A) is non-zero, calculate the inverse using:\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \], where \( \text{adj}(A) \) is the adjugate of \( A \).

Step 3: Solve for Variable Vector

Multiply the inverse matrix \( A^{-1} \) by the constant matrix \( \mathbf{b} \) to solve for \( \mathbf{x} \):\[ \mathbf{x} = A^{-1} \cdot \mathbf{b} \]Perform matrix multiplication to find the values of \( x \), \( y \), and \( z \). Ensure each step is calculated accurately to get the correct solution set \( (x, y, z) \).

Step 4: Verify the Solution

Substitute the values of \( x \), \( y \), and \( z \) obtained back into the original equations to verify correctness. Each equation should satisfy the left-hand side equaling the right-hand side within acceptable precision. This double-checks calculations and confirms the solution.

Key Concepts

Matrix Inversion

Matrix inversion is an essential process in solving systems of linear equations, especially when using a matrix-based approach. Essentially, matrix inversion involves finding a matrix that can "undo" the multiplication of another matrix. For a matrix \( A \), its inverse \( A^{-1} \) is such that when you multiply \( A \) by \( A^{-1} \), the result is the identity matrix \( I \):\[ A \cdot A^{-1} = I \]The identity matrix is like "1" in matrix algebra. It does not change another matrix when multiplied by it. Not all matrices have an inverse, though. A matrix must be square (same number of rows and columns) and have a non-zero determinant.To find the inverse, we often calculate the adjugate of the matrix and use it with the reciprocal of the determinant:- Calculate the determinant of matrix \( A \). If it is zero, then \( A \) does not have an inverse.- Compute the adjugate matrix (\( \text{adj}(A) \)).- Use the formula:\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]Inversions are key in solving matrix equations because they allow us to isolate variables effectively by transforming the coefficient matrix.

System of Linear Equations

A system of linear equations is a collection of equations where all terms are either constant or the product of a constant and a variable. They appear as straight lines in a geometric sense. These systems can be solved using various methods, one of which is the matrix method.For example, consider the system given:\[\begin{align*}4x + 3y - z &= 0 \3x + 2y + 5z &= 6 \5x - y - 3z &= 3\end{align*}\]Using matrices, we convert this to a matrix equation \( A \cdot \mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) are the variables, and \( \mathbf{b} \) holds the constants.To solve:1. Construct the coefficient matrix (\( A \)) using the coefficients of the equations.2. Use matrix inversion (if possible) to find the vector of variables \( \mathbf{x} \).The goal is to simplify the problem of multiple equations into a single matrix equation, making solutions systematic and straightforward.

Determinant of a Matrix

The determinant of a matrix is a unique number that can be calculated from its elements, and it provides important information about the matrix. For a square matrix \( A \), its determinant (often denoted \( \text{det}(A) \)) is crucial in many areas, such as calculating the inverse of the matrix.For example, the determinant helps in:- Checking if a matrix has an inverse. If \( \text{det}(A) = 0 \), the matrix does not have an inverse.- Determining the volume scaling factor of the transformation characterized by the matrix.To manually calculate a determinant for a 3x3 matrix, use the formula:\[\text{det}(A) = a(ei−fh) − b(di−fg) + c(dh−eg)\]where \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \).Understanding the determinant is foundational as it ensures the matrix operations we perform lead to valid solutions, especially when solving systems of equations.