## MATRIX Course

# Cofactor expansion

When we have a big matrix, we can split the matrix into smaller matrices, and use formulas we know such as $ad-bc$ for the 2x2 matrix. In French, our teacher is also calling this `Calcul par développement`

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We are calling **minor** of a matrix $A$, the matrix $M_{i,p}$ created by removing the row $i$ and the column $p$ of $A$.

We are calling **cofactor** $C_{i,p} = (-1)^{i+p} * det(M_{i,p})$.

## Formula

We will pick a column $p$ and remove it from our matrix $A$. It's most of the time the column with the biggest coefficients or the coefficients that are hard to deal with. Then, all we have to do is to evaluate this formula.

*Note*: this is a sum of the product of the coefficient of the column we removed by their cofactors.

## Example

We are picking the column $p=3 \to (-7,8,-9)$. The formula is

- $C_{1,3} = (-1)^{4} * det(M_{1,3}) = det(M_{1,3}) = \textbf{-3}$

- $C_{2,3} = (-1)^{5} * det(M_{2,3}) = -det(M_{2,3}) = \textbf{-6}$

- $C_{3,3} = (-1)^{6} * det(M_{3,3}) = det(M_{3,3}) = \textbf{-3}$

Hence, the result is