Determinant of a 3×3 Matrix via Cofactor Expansion

Cofactor expansion — the recipe

For any row $i$ (or column $j$) of an $n \times n$ matrix $A$: $\det A = \sum_{j=1}^{n} (-1)^{i+j} A_{ij} M_{ij}$

where $M_{ij}$ is the minor — the determinant of the $(n-1) \times (n-1)$ submatrix obtained by deleting row $i$ and column $j$.

The sign factor $(-1)^{i+j}$ follows the checkerboard pattern: $\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$

Step-by-step — expand along row 1

$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{pmatrix}$

Step 1 — Minor $M_{11}$: delete row 1 and column 1. $M_{11} = \det \begin{pmatrix} 5 & 6 \\ 8 & 0 \end{pmatrix} = (5)(0) - (6)(8) = -48$

Step 2 — Minor $M_{12}$: delete row 1 and column 2. $M_{12} = \det \begin{pmatrix} 4 & 6 \\ 7 & 0 \end{pmatrix} = (4)(0) - (6)(7) = -42$

Step 3 — Minor $M_{13}$: delete row 1 and column 3. $M_{13} = \det \begin{pmatrix} 4 & 5 \\ 7 & 8 \end{pmatrix} = (4)(8) - (5)(7) = 32 - 35 = -3$

Step 4 — Combine with signs: $\det A = (+1)(1)(-48) + (-1)(2)(-42) + (+1)(3)(-3)$ $= -48 + 84 - 9$ $= \boxed{27}$

Choosing the best row or column

Cofactor expansion works along any row or column. Pick the one with the most zeros to save work. If a row is $(0, 0, 5)$, only one of its three minors contributes — huge time saver.

Row reduction as a faster alternative

For matrices $\ge 3 \times 3$, row reducing to upper triangular form is often faster. The determinant of an upper triangular matrix is the product of its diagonal entries. Track row operation effects:

• Row swap: flip sign of running determinant.
• Scale row by $c$: multiply running determinant by $c$.
• Add multiple of a row to another: no change to determinant.

Properties (reprise)

• $\det A = 0$ iff rows (or columns) are linearly dependent.
• $\det(A^T) = \det A$ — so you can always expand along a column too.
• $\det(AB) = \det(A) \det(B)$.
• $\det(A^n) = (\det A)^n$.

Common mistakes

• Dropping the sign pattern. The $(-1)^{i+j}$ factor is essential: $+, -, +, -, \ldots$
• Computing minors wrong. To delete row $i$, column $j$, cross them both out and use what remains.
• Expanding along a random row. Pick a row or column with zeros — much less arithmetic.

Try it in the visualization

Each of the three $2 \times 2$ minors is highlighted in turn as you step through the expansion. Signs alternate, values are shown, and the running sum converges to the final determinant.