To calculate the determinant of a 3 by 3 matrix, we build upon the method used for 2 by 2 matrices, which involves multiplying the diagonals and subtracting. However, the process for 3 by 3 matrices is slightly more complex, as it requires calculating several 2 by 2 determinants.
For a 3 by 3 matrix represented as:
\[\begin{bmatrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{bmatrix}\]
the determinant can be calculated using the formula:
\[\text{det}(A) = a_{11} \cdot \text{det}\begin{bmatrix}a_{22} & a_{23} \\a_{32} & a_{33}\end{bmatrix} - a_{12} \cdot \text{det}\begin{bmatrix}a_{21} & a_{23} \\a_{31} & a_{33}\end{bmatrix} + a_{13} \cdot \text{det}\begin{bmatrix}a_{21} & a_{22} \\a_{31} & a_{32}\end{bmatrix}\]
In this formula, the signs alternate: the first term is positive, the second is negative, and the third is positive. This pattern is crucial for correctly calculating the determinant.
To illustrate this, consider the matrix:
\[\begin{bmatrix}3 & 1 & 0 \\2 & -3 & -1 \\4 & 6 & 0\end{bmatrix}\]
We start by calculating the determinant using the first row:
- For the first term, take \(3\) and calculate the determinant of the 2 by 2 matrix formed by eliminating the first row and first column:
- For the second term, take \(1\) and calculate the determinant of the 2 by 2 matrix formed by eliminating the first row and second column:
- For the third term, take \(0\) and calculate the determinant of the 2 by 2 matrix formed by eliminating the first row and third column:
\[\text{det}\begin{bmatrix}-3 & -1 \\6 & 0\end{bmatrix} = (-3) \cdot 0 - (-1) \cdot 6 = 6\]
\[\text{det}\begin{bmatrix}2 & -1 \\4 & 0\end{bmatrix} = (2) \cdot 0 - (-1) \cdot 4 = 4\]
\[\text{det}\begin{bmatrix}2 & -3 \\4 & 6\end{bmatrix} = (2) \cdot 6 - (-3) \cdot 4 = 12 + 12 = 24\]
Now, substituting these values back into the determinant formula gives:
\[\text{det}(A) = 3 \cdot 6 - 1 \cdot 4 + 0 \cdot 24 = 18 - 4 + 0 = 14\]
Thus, the determinant of the 3 by 3 matrix is \(14\). This method effectively reduces the complexity of calculating a 3 by 3 determinant by breaking it down into manageable 2 by 2 determinants, allowing for a systematic approach to finding the solution.
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