How to Calculate the Determinant of a 3x3 Matrix

To calculate the determinant of a 3x3 matrix, you can follow these steps:

Step 1: Write down the matrix. For example, let's say we have a 3x3 matrix A:

A = | a b c | | d e f | | g h i |

Step 2: Use the Rule of Sarrus or the cofactor expansion method. I'll explain the cofactor expansion method here.

Step 3: Choose any row or column for expansion. For simplicity, let's choose the first row.

Step 4: For each element in the row, multiply it by the determinant of the 2x2 submatrix that is left when you remove the row and column of that element. Also, alternate signs, starting with a plus for the first element.

The determinant (Det) of matrix A is:

Det(A) = a * Det(Minor of a) - b * Det(Minor of b) + c * Det(Minor of c)

Here, the "Minor of a" is the 2x2 matrix you get after removing the row and column of 'a':

Minor of a = | e f | | h i |

The determinant of the Minor of 'a' (a 2x2 matrix) is calculated as:

Det(Minor of a) = (e*i) - (f*h)

Similarly, calculate the determinant for the Minor of 'b' and Minor of 'c'.

Step 5: Substitute the values and solve.

For example, if the matrix A is:

| 1 2 3 | | 4 5 6 | | 7 8 9 |

Then the determinant is:

Det(A) = 1*((5*9)-(6*8)) - 2*((4*9)-(6*7)) + 3*((4*8)-(5*7)) = 1*(-3) - 2*(-6) + 3*(-4) = -3 + 12 - 12 = -3

So the determinant of A is -3.