Determinant of a 3x3 Matrix

📌 CONCEPT: A determinant of a 3x3 matrix A is zero if at least one row or column is a linear combination of other rows or columns, meaning the rows or columns are not linearly independent.

📐 RULE / FORMULA: To calculate the determinant of a 3x3 matrix, we use the formula: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg), where the elements of the matrix are denoted by their respective positions.

💡 WORKED EXAMPLE: Consider the 3x3 matrix A = | 1^2 2^2 3^2 | | 4^2 5^2 6^2 | | 7^2 8^2 9^2 |

Using the formula, we get det(A) = 1(5^2*9 - 6^2*8) - 2(4^2*9 - 6^2*7) + 3(4^2*8 - 5^2*7) = 0.

⚠️ COMMON MISTAKE: Students often get confused in expanding the determinant using the formula, so it's essential to carefully calculate each term and follow the order of operations to avoid errors.