Lesson 3.2: Calculation and Properties of the Determinant

The machinery for computing and reasoning about the scaling factor of space.

In our last lesson, we developed a deep geometric intuition for the determinant. Now, it's time to build the machinery to compute this magical number for any square matrix.

The 3x3 Case: Cofactor Expansion

This method breaks down the determinant of an `n x n` matrix into a combination of determinants of smaller `(n-1) x (n-1)` matrices.

For a general 3x3 matrix expanded along the first row, the formula is:

\det(A) = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix}

Practical Tip

Always choose to expand along the row or column with the most zeros! This minimizes the number of cofactors you need to calculate.

Key Properties of the Determinant

1. Identity Matrix: $\det(I) = 1$ .

2. Row Swaps: Swapping two rows flips the sign of the determinant.

3. Row Replacement: Adding a multiple of one row to another does not change the determinant.

4. Triangular Matrix: The determinant is the product of the diagonal entries.

5. Invertibility: $A$ is invertible if and only if $\det(A) \neq 0$ .

6. Multiplicative Property: $\det(AB) = \det(A) \cdot \det(B)$ .

7. Inverse: $\det(A^{-1}) = 1 / \det(A)$ .

8. Transpose: $\det(A^T) = \det(A)$ .

A Smarter Way to Calculate

Brute-force cofactor expansion is very slow for large matrices. The most efficient way to calculate a determinant is to use row operations to get to REF, then multiply the pivots.

Step 1: Use Row Operations to get to REF

Given $A = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \\ 1 & 2 & 4 \end{bmatrix}$ , factor 2 from R1, then perform elimination.

$\det(A) = 2 \cdot \det(\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix})$

Step 2: Multiply the Diagonals of U and the factored scalars

The REF matrix `U` has a determinant of 1 (1x1x1).

The final determinant is $\det(A) = 2 \times 1 = 2$ . This is far faster than a full cofactor expansion.

Lesson 3.1: The Geometric Meaning of the Determinant

Lesson 4.1: Eigenvalues & Eigenvectors