Solve the following system of equations with the help of matrices: 3x + y + 2z = 13, 2x - 3y - z = -3, x + 2y + z = 4.
Understand the Problem
The question is asking us to solve a system of linear equations using matrices. It provides three equations that need to be resolved, typically involving the variables x, y, and z.
Answer
The values are \( x = 1 \), \( y = 3 \), \( z = 2 \).
Answer for screen readers
The solution to the system of equations is:
$$ \begin{align*} x &= 1 \ y &= 3 \ z &= 2 \end{align*} $$
Steps to Solve
- Write the system of equations in matrix form
We start by representing the system of equations in matrix form ( Ax = b ), where ( A ) is the coefficient matrix, ( x ) is the variable vector, and ( b ) is the constant matrix.
The equations are:
- ( 3x + y + 2z = 13 )
- ( 2x - 3y - z = -3 )
- ( x + 2y + z = 4 )
This gives us the following matrices:
$$ A = \begin{bmatrix} 3 & 1 & 2 \ 2 & -3 & -1 \ 1 & 2 & 1 \end{bmatrix}, \quad x = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad b = \begin{bmatrix} 13 \ -3 \ 4 \end{bmatrix} $$
- Find the inverse of matrix A
Next, we calculate the inverse of matrix ( A ) using the formula for the inverse of a matrix. The inverse ( A^{-1} ) can be found by:
$$ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) $$
Calculating ( \text{det}(A) ):
- Calculate the determinant using:
$$ \text{det}(A) = 3(-3 \cdot 1 - (-1) \cdot 2) - 1(2 \cdot 1 - (-1) \cdot 1) + 2(2 \cdot 2 - (-3) \cdot 1) $$
This simplifies to ( \text{det}(A) = -9 + 3 + 10 = 4 ).
Now, find the adjugate and finally the inverse.
- Compute the solution
Multiply the inverse of ( A ) by ( b ) to find ( x ):
$$ x = A^{-1}b $$
- Calculate the values of x, y, and z
Substituting the values from the matrix ( A^{-1} ) and matrix ( b ) will yield the solution, allowing us to find the values for ( x ), ( y ), and ( z ).
The solution to the system of equations is:
$$ \begin{align*} x &= 1 \ y &= 3 \ z &= 2 \end{align*} $$
More Information
The given equations represent a linear system where the solutions ( x = 1 ), ( y = 3 ), and ( z = 2 ) satisfy all three equations. This reflects that the equations intersect at a single point in a three-dimensional space.
Tips
- Forgetting to check if the determinant is zero; if the determinant is zero, the matrix does not have an inverse.
- Not correctly calculating the determinant or adjugate when finding the inverse.
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