QuestionAugust 27, 2025

Solve the following system of equations by using matrix inverse method. 2x-y+3z=9 x+y+z=6 x-y+z=2 x=1,y=1 , and z=3 x=1,y=2 and z=6 x=2,y=2 and z=9 x=1,y=2 and z=3

Solve the following system of equations by using matrix inverse method. 2x-y+3z=9 x+y+z=6 x-y+z=2 x=1,y=1 , and z=3 x=1,y=2 and z=6 x=2,y=2 and z=9 x=1,y=2 and z=3
Solve the following system of equations by using matrix inverse method. 2x-y+3z=9 x+y+z=6 x-y+z=2 x=1,y=1 , and z=3 x=1,y=2 and z=6 x=2,y=2 and z=9 x=1,y=2 and z=3

Solution
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Answer

x=1, y=2, z=3 Explanation 1. Write the system in matrix form The system can be written as AX = B, where A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 1 & 1 \\ 1 & -1 & 1 \end{bmatrix}, X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, and B = \begin{bmatrix} 9 \\ 6 \\ 2 \end{bmatrix}. 2. Calculate the inverse of matrix A Find A^{-1} using the formula for the inverse of a 3x3 matrix. After calculation, A^{-1} = \begin{bmatrix} \frac{1}{2} & 0 & \frac{1}{2} \\ 1 & 1 & -2 \\ -\frac{3}{2} & -1 & \frac{5}{2} \end{bmatrix}. 3. Multiply the inverse by B Compute X = A^{-1}B. Perform the matrix multiplication to find X = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}.

Explanation

1. Write the system in matrix form<br /> The system can be written as $AX = B$, where $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 1 & 1 \\ 1 & -1 & 1 \end{bmatrix}$, $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$, and $B = \begin{bmatrix} 9 \\ 6 \\ 2 \end{bmatrix}$.<br />2. Calculate the inverse of matrix A<br /> Find $A^{-1}$ using the formula for the inverse of a 3x3 matrix. After calculation, $A^{-1} = \begin{bmatrix} \frac{1}{2} & 0 & \frac{1}{2} \\ 1 & 1 & -2 \\ -\frac{3}{2} & -1 & \frac{5}{2} \end{bmatrix}$.<br />3. Multiply the inverse by B<br /> Compute $X = A^{-1}B$. Perform the matrix multiplication to find $X = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$.
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