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When you first learn to evaluate a 3x3 matrix, you often just memorise the formula for expanding along the top row. But determinants are much more flexible than that. Underneath that top-row formula are two mathematical ideas: minors and cofactors.
What are minors and cofactors?
A minor of an element in a matrix is simply a smaller determinant. If you pick an element in row i and column j, its minor (written as M_ij) is the determinant of the submatrix left over when you delete that entire row and column (Wikipedia)).
A cofactor (C_ij) is the exact same number, but with a specific sign attached. You find the cofactor by multiplying the minor by (−1)^(i+j).
C_ij = (−1)^(i+j) × M_ij
How to find minors and cofactors of a 3x3 matrix?
Let's break down the steps for a 3x3 matrix. Suppose you want to find the minor M_12 (row 1, column 2):
Cross out the entire first row and the second column.
You are left with a 2x2 matrix.
Find its determinant by multiplying the diagonals and subtracting them: (a × d) − (b × c).
To get the cofactor C_12, you add the row and column numbers together: 1 + 2 = 3. Because 3 is an odd number, (−1)³ = −1. Therefore, the cofactor is the negative of the minor: C_12 = −M_12.
If keeping track of these steps feels overwhelming, Fermi's AI tutor can build a plan for your study session. When you are stuck on a matrix problem, it walks you through finding the determinant step by step, checking your signs rather than just printing the final answer.
How does the (−1)^(i+j) sign checkerboard work?
Instead of calculating (−1)^(i+j) every single time, you can picture a sign checkerboard. Because the sum of the row and column numbers goes up by one at each horizontal or vertical step, the signs alternate. For a 3x3 matrix, the pattern is always:
+ − +
− + −
+ − +
This pattern means you do not have to stick to the top row when calculating a determinant. You can expand along any row or column. If you spot a row with a zero, expand along that one to save time. Just pick the elements of that row, multiply them by their corresponding cofactors, and add them up.
If you need a refresher on where else these patterns show up, such as finding Aⁿ for a matrix by spotting the pattern or calculating the area of a triangle using determinants, you will see the same rules applying again and again.
Why does a row times another row's cofactors give zero?
When you multiply a row's elements by their *own* cofactors and add them up, you get the determinant of the matrix. But what happens if you multiply the elements of row 1 by the cofactors of row 2?
You get zero. Always.
This is sometimes called the "alien cofactor" property. Mathematically, multiplying row 1's elements by row 2's cofactors is exactly the same as finding the determinant of a new matrix where row 2 has been replaced by a copy of row 1. Since any matrix with two identical rows has a determinant of zero (Wikipedia), the sum vanishes.
This rule might feel like a trick, but it is the engine that makes matrix inverses work. The entire theory of linear algebra is built on these hidden structures, all mapped out in the Concept Library.
