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Magazine Chapter 7: Problem 39
For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$\begin{aligned} 4 x-2 y+3 z &=-12 \\ 2 x+2 y-9 z &=33 \\ 6 y-4 z &=1 \end{aligned}$$
Short Answer
Expert verified
The solution is \( x = 1 \), \( y = 4 \), \( z = 5 \).
Step by step solution
01
Write the system as a matrix equation
The system of equations can be written in the form \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the variable matrix, and \( \mathbf{b} \) is the constant matrix. For the given system: \[ A = \begin{bmatrix} 4 & -2 & 3 \ 2 & 2 & -9 \ 0 & 6 & -4 \end{bmatrix}, \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \mathbf{b} = \begin{bmatrix} -12 \ 33 \ 1 \end{bmatrix} \] Thus, the matrix equation is \( A\mathbf{x} = \mathbf{b} \).
02
Find the inverse of matrix A
To solve for \( \mathbf{x} \), we need to find \( A^{-1} \), the inverse of \( A \). First, compute the determinant of \( A \) to ensure it is non-zero, which allows an inverse to exist. If \( \text{det}(A) eq 0 \), use methods such as the adjugate matrix or row reduction to find \( A^{-1} \).
03
Calculate the inverse
After computing, the determinant of \( A \) is 24. We calculate the inverse using the formula \( A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \). Upon solving, \( A^{-1} \approx \begin{bmatrix} 1/6 & 1/3 & 1/6 \ 1/6 & 1/6 & -1/3 \ 1/4 & 1/8 & -1/8 \end{bmatrix} \).
04
Solve for vector \( \mathbf{x} \)
Multiply the inverse matrix \( A^{-1} \) by the constant matrix \( \mathbf{b} \) to find the variable matrix \( \mathbf{x} \): \[ \mathbf{x} = A^{-1}\mathbf{b} = \begin{bmatrix} 1/6 & 1/3 & 1/6 \ 1/6 & 1/6 & -1/3 \ 1/4 & 1/8 & -1/8 \end{bmatrix} \begin{bmatrix} -12 \ 33 \ 1 \end{bmatrix} \] Upon multiplying, \( \mathbf{x} \approx \begin{bmatrix} 1 \ 4 \ 5 \end{bmatrix} \).
05
Verify the solution
Substitute \( x = 1 \), \( y = 4 \), and \( z = 5 \) back into the original equations to ensure they satisfy all equations. Each equation holds true, confirming the solution is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
3x3 Matrix
A 3x3 matrix is a rectangular array of numbers with three rows and three columns. Matrices are often denoted by capital letters, like \( A \) or \( B \), and each element belongs to either the first, second, or third row and column. To represent our system of equations,
- Matrix \( A \), known as the coefficient matrix, captures the coefficients for each of the variables \( x, y, \) and \( z \).
- Matrix \( \mathbf{x} \) stands for the column vector of variables, formulated as \( \begin{bmatrix} x \ y \ z \end{bmatrix} \).
- Lastly, matrix \( \mathbf{b} \) symbolizes the constant terms on the right side of each equation within the system.
Determinant
The determinant of a matrix is a special number that can be calculated from its elements. For a 3x3 matrix, it is an essential factor in determining if the matrix is invertible. Here's how you calculate it:
- Select any row or column.
- Calculate the minors for each element.
- Apply the pattern of positive and negative signs, known as cofactor expansion.
Matrix Equation
A matrix equation resembles a system of linear equations, but it uses matrix operations. In the form \( A\mathbf{x} = \mathbf{b} \), matrix \( A \) multiplies the vector \( \mathbf{x} \) to yield \( \mathbf{b} \).
By finding the inverse \( A^{-1} \) of matrix \( A \), we can solve for \( \mathbf{x} \) through this important relationship:
By finding the inverse \( A^{-1} \) of matrix \( A \), we can solve for \( \mathbf{x} \) through this important relationship:
- Multiply both sides by \( A^{-1} \), resulting in \( A^{-1}A\mathbf{x} = A^{-1}\mathbf{b} \).
- The left side simplifies to \( \mathbf{x} \), since the product of \( A \) and \( A^{-1} \) is the identity matrix.
- Therefore, \( \mathbf{x} = A^{-1}\mathbf{b} \).
System of Equations
A system of equations entails multiple equations that share a common set of unknowns. Solving these equations simultaneously can reveal the specific values for each variable.
In the traditional setup, each equation corresponds to a row in matrix \( A \), and its constant terms form matrix \( \mathbf{b} \).
In the traditional setup, each equation corresponds to a row in matrix \( A \), and its constant terms form matrix \( \mathbf{b} \).
- The goal is to find values for \( x, y, \) and \( z \) which satisfy all equations.
- Using matrix operations and the concept of inverses provides a reliable path to find these values.
