The determinant of a 2x2 matrix:
The inverse of a matrix only exists where
- There is a matrix minor corresponding to each element of a matrix
- The minor is calculated by
- ignoring the values on the current row and column
- calculate the determinant of the remaining 2x2 matrix
The minor of the top left corner is:
The cofactor is the minor multiplied by it's correct sign. The signs form a checkerboard pattern:
The matrix of cofactors is denoted .
The determinant of a 3x3 matrix is calculated by multiplying each element in one row/column by it's cofactor, then summing them. For the matrix:
This shows the expansion of the top row, but any column or row will produce the same result.
- Calculate matrix of minors
- Calculate matrix of cofactors
- Multiply by 1 over determinant
The transposed matrix of cofactors is therefore:
Explanding by the bottom row to calculate the determinant (it has 2 zeros so easy calculation):