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Second method for 3 by 3 Determinants (#19090) 

Second method for 3 by 3 matrix Determinants, based on extended matrix diagonals
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title: 3 by 3 Determinants title: 3 by 3 Determinants
--- ---
## 3 by 3 Determinants ## 3 by 3 Determinants
### Method 1
Consider the following matrix, which we will call A: Consider the following matrix, which we will call A:
<table> <table>
@ -57,6 +57,171 @@ For example, consider the following matrix, which we will call B:
<em>det(B) = (-41) - 100 - 90 = -231</em> <em>det(B) = (-41) - 100 - 90 = -231</em>
### Method 2
This method it similar to 2 by 2 determinants, and based on opertations with diagonals
Again, consider the following matrix, which we will call A:
<table>
<tr>
<td style="background-color: white">a</td>
<td style="background-color: white">b</td>
<td style="background-color: white">c</td>
</tr>
<tr>
<td style="background-color: white">d</td>
<td style="background-color: white">e</td>
<td style="background-color: white">f</td>
</tr>
<tr>
<td style="background-color: white">g</td>
<td style="background-color: white">h</td>
<td style="background-color: white">i</td>
</tr>
</table>
Then the determinant of this matrix, denoted <em>det(A)</em>, is given by:
<em>det(A) = a * e * i + b * f * g + c * d * h - c * e * g - f * h * a - i * b * d</em>
Note how three top-right to bottom-left diagonals are positive
<table>
<tr>
<td style="background-color: white">a</td>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
</tr>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white">e</td>
<td style="background-color: white"> </td>
</tr>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
<td style="background-color: white">i</td>
</tr>
</table>
<table>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white">b</td>
<td style="background-color: white"> </td>
</tr>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
<td style="background-color: white">f</td>
</tr>
<tr>
<td style="background-color: white">g</td>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
</tr>
</table>
<table>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
<td style="background-color: white">c</td>
</tr>
<tr>
<td style="background-color: white">d</td>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
</tr>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white">h</td>
<td style="background-color: white"> </td>
</tr>
</table>
Top-left to bottom-right are negative
<table>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
<td style="background-color: white">c</td>
</tr>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white">e</td>
<td style="background-color: white"> </td>
</tr>
<tr>
<td style="background-color: white">g</td>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
</tr>
</table>
<table>
<tr>
<td style="background-color: white">a</td>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
</tr>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
<td style="background-color: white">f</td>
</tr>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white">h</td>
<td style="background-color: white"> </td>
</tr>
</table>
<table>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white">b</td>
<td style="background-color: white"> </td>
</tr>
<tr>
<td style="background-color: white">d</td>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
</tr>
<tr>
<td style="background-color: white"> </td>
<td style="background-color: white"> </td>
<td style="background-color: white">i</td>
</tr>
</table>
Consider the same example as in method 2: matrix, which we will call B:
<table>
<tr>
<td style="background-color: white">1</td>
<td style="background-color: white">2</td>
<td style="background-color: white">3</td>
</tr>
<tr>
<td style="background-color: white">0</td>
<td style="background-color: white">-3</td>
<td style="background-color: white">5</td>
</tr>
<tr>
<td style="background-color: white">-10</td>
<td style="background-color: white">4</td>
<td style="background-color: white">7</td>
</tr>
</table>
<em>det(B)</em> is given by the formula above. We apply the formula below:
<em>det(B) = 1 * (-3) * 7 + 2 * 5 * (-10) + 3 * 0 * 4 - 3 * (-3) * (-10) - 5 * 4 * 1 - 7 * 2 * 0</em>, which we simplify to:
<em>det(B) = -21 - 100 + 0 - 90 - 20 - 0 = -231</em>, same, as in method 1
#### More information: #### More information:
* [Determinant of a Matrix](https://www.mathsisfun.com/algebra/matrix-determinant.html) on MathIsFun * [Determinant of a Matrix](https://www.mathsisfun.com/algebra/matrix-determinant.html) on MathIsFun
* [3x3 Determinant calculator](http://www.wolframalpha.com/widgets/view.jsp?id=7fcb0a2c0f0f41d9f4454ac2d8ed7ad6) * [3x3 Determinant calculator](http://www.wolframalpha.com/widgets/view.jsp?id=7fcb0a2c0f0f41d9f4454ac2d8ed7ad6)