Solve the equations by matrix inversion method: - 
x + y + z = 3;               x + 2y + 3z = 4 ;        x + 4y + 9z = 6  

The given system of linear equations can be represented as:

\[
\begin{aligned}
1x + 1y + 1z &= 3 \\
1x + 2y + 3z &= 4 \\
1x + 4y + 9z &= 6
\end{aligned}
\]

This system of equations can be written in matrix form as:

\[
A \cdot X = B
\]

Where:
\[
A = \begin{bmatrix} 
1 & 1 & 1 \\ 
1 & 2 & 3 \\ 
1 & 4 & 9 
\end{bmatrix},
\quad X = \begin{bmatrix} 
x \\ 
y \\ 
z 
\end{bmatrix},
\quad B = \begin{bmatrix} 
3 \\ 
4 \\ 
6 
\end{bmatrix}
\]

### Step 1: Find the inverse of matrix \( A \)

To solve for \( X \), we need to calculate:

\[
X = A^{-1} \cdot B
\]

We'll compute \( A^{-1} \) (the inverse of matrix \( A \)) using standard methods like the adjugate formula or row reduction.

### Step 2: Compute determinant of \( A \)

The determinant of matrix \( A \) is:

\[
\text{det}(A) = 1 \times (2 \times 9 - 4 \times 3) - 1 \times (1 \times 9 - 1 \times 3) + 1 \times (1 \times 4 - 1 \times 2)
\]
\[
\text{det}(A) = 1 \times (18 - 12) - 1 \times (9 - 3) + 1 \times (4 - 2) = 6 - 6 + 2 = 2
\]

Since \( \text{det}(A) = 2 \), the matrix is invertible.

### Step 3: Find the cofactor matrix of \( A \)

Next, we find the cofactor matrix, which is the matrix of minors with signs:

\[
\text{Cofactor}(A) = \begin{bmatrix} 
2 & -6 & 4 \\ 
-6 & 8 & -2 \\ 
4 & -2 & 0 
\end{bmatrix}
\]

### Step 4: Find the adjugate of \( A \)

The adjugate of \( A \) is the transpose of the cofactor matrix:

\[
\text{Adj}(A) = \begin{bmatrix} 
2 & -6 & 4 \\ 
-6 & 8 & -2 \\ 
4 & -2 & 0 
\end{bmatrix}^T = \begin{bmatrix} 
2 & -6 & 4 \\ 
-6 & 8 & -2 \\ 
4 & -2 & 0 
\end{bmatrix}
\]

### Step 5: Compute the inverse of \( A \)

The inverse of \( A \) is:

\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{Adj}(A) = \frac{1}{2} \cdot \begin{bmatrix} 
2 & -6 & 4 \\ 
-6 & 8 & -2 \\ 
4 & -2 & 0 
\end{bmatrix}
\]
\[
A^{-1} = \begin{bmatrix} 
1 & -3 & 2 \\ 
-3 & 4 & -1 \\ 
2 & -1 & 0 
\end{bmatrix}
\]

### Step 6: Multiply \( A^{-1} \) with \( B \)

Now, we multiply \( A^{-1} \) with \( B \) to find \( X \):

\[
X = A^{-1} \cdot B = \begin{bmatrix} 
1 & -3 & 2 \\ 
-3 & 4 & -1 \\ 
2 & -1 & 0 
\end{bmatrix} \cdot \begin{bmatrix} 
3 \\ 
4 \\ 
6 
\end{bmatrix}
\]

Perform the matrix multiplication:

\[
X = \begin{bmatrix} 
1 \times 3 + (-3) \times 4 + 2 \times 6 \\ 
-3 \times 3 + 4 \times 4 + (-1) \times 6 \\ 
2 \times 3 + (-1) \times 4 + 0 \times 6 
\end{bmatrix}
\]
\[
X = \begin{bmatrix} 
3 - 12 + 12 \\ 
-9 + 16 - 6 \\ 
6 - 4 + 0 
\end{bmatrix} = \begin{bmatrix} 
3 \\ 
1 \\ 
2 
\end{bmatrix}
\]

### Solution:

\[
x = 3, \quad y = 1, \quad z = 2
\]

Would you like a deeper explanation of any step, or have any questions?

Here are five related problems you might consider:
1. Solve a 3x3 system using Gaussian elimination.
2. Find the inverse of a 3x3 matrix without using determinants.
3. Solve a system of equations using Cramer's rule.
4. Solve a system of nonlinear equations using matrix methods.
5. Analyze the solution of a 2x2 matrix system with a singular matrix.

**Tip:** When solving systems of equations, checking the determinant first can quickly indicate whether the matrix is invertible.

Solve the equations by matrix inversion method: x + y + z = 3; x + 2y + 3z = 4; x + 4y + 9z = 6

Learn how to solve the system of linear equations x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6 using the matrix inversion method. This detailed guide covers matrix representation, determinant calculation, and matrix multiplication for Grades 10-12 and undergraduates.

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