Let the matrix be
[
begin{pmatrix}
a & b & c \
d & e & f \
g & h & i
end{pmatrix}
]
Use the formula
[
det = a(ei – fh) – b(di – fg) + c(dh – eg)
]
Multiply (a) by the determinant of the (2 times 2) matrix left after removing row 1 and column 1
Subtract (b) times the determinant of the (2 times 2) matrix left after removing row 1 and column 2
Add (c) times the determinant of the (2 times 2) matrix left after removing row 1 and column 3
Compute the (2 times 2) determinants:
[
ei – fh,quad di – fg,quad dh – eg
]
Combine the results:
[
det = a(ei – fh) – b(di – fg) + c(dh – eg)
]