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. We know it's the scaling factor of a transformation. Now, it's time to build the machinery to compute this magical number for any square matrix.

We'll start with the simple cases and build up to a general, powerful formula. Then, we'll uncover the properties that let us reason about determinants without always resorting to heavy computation.

The 2x2 Case: The Formula We Already Know

For a 2x2 matrix, the formula is simple and is the one we discovered geometrically.

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \implies \det(A) = ad - bc

This formula represents the signed area of the parallelogram formed by the transformed basis vectors $[a, c]$ and $[b, d]$ .

Example:

A = \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix}

$\det(A) = (3)(4) - (-1)(2) = 12 - (-2) = 14$ . This transformation scales area by a factor of 14 and preserves orientation.

The 3x3 Case: Introducing 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 \cdot C_{11} + b \cdot C_{12} + c \cdot C_{13}

The $C_{ij}$ terms are **cofactors**, where $C_{ij} = (-1)^{i+j} \cdot \det(M_{ij})$ . $M_{ij}$ is the **minor**, the smaller matrix left when you delete row `i` and column `j`. The $(-1)^{i+j}$ term creates a checkerboard pattern of signs:

\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}

Putting it all together:

\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}

= a(ei - fh) - b(di - fg) + c(dh - eg)

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: $\det(I) = 1$ . (The "do-nothing" transformation has a scaling factor of 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. This is why Gaussian elimination is so useful!

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: $\det(AB) = \det(A) \cdot \det(B)$ . The scaling factor of composed transformations is the product of their individual scaling factors.

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 combine row operations with the triangular matrix property.

To find $\det(A)$ for $A = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \\ 1 & 2 & 4 \end{bmatrix}$ :

Use row operations to get to REF, keeping track of changes.
- Factor 2 from R1: $\det(A) = 2 \cdot \det(\begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 5 \\ 1 & 2 & 4 \end{bmatrix})$
- $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$ (no change to determinant).
The matrix becomes upper triangular: $U = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}$ . Its determinant is the product of the diagonals: $1 \times 1 \times 1 = 1$ .
The final determinant is $\det(A) = 2 \times 1 = 2$ . This is far faster than a full cofactor expansion.

**Up Next in Module 4:** Armed with the power of the determinant, we are ready to tackle the central topic of the second half of linear algebra: **Eigenvalues and Eigenvectors**.