math111_logo Proof of Properties

2. Justification of computation

We prove the two computational tools for the determinant: row/column operations and cofactor expansions.

Column Operation The properties, stated for columns, are

A changed to B by determinant changed by
r[col i] + [col j] detB = detA
[col i] ↔ [col j] detB = - detA
d[col i], d ≠ 0 detB = d detA

For the first operation, we have

det(..., u, ..., ru + v, ...)
= r det(..., u, ..., u, ...) + det(..., u, ..., v, ...) (det is multilinear)
= det(..., u, ..., v, ...) (this consequence of alternating property)

For the second operation, the property is exactly the alternating property, which is one of the defining properties of det.

For the third operation, the property is the (scalar part of) multilinearity property, which is also built into the definition of det.

Row Operation This follows from the column operation case and the property detA = detAT proved before.

Cofactor Expansion Along Column Let us consider the cofactor expansion along the first column of a 4 by 4 matrix

A = [a1 a2 a3 a4] = [ a11 a12 a13 a14 ].
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44

By the linearity in the first column, we have

detA = det(a11e1 + a21e2 + a31e3 + a41e4, a2, a3, a4)
= a11det(e1, a2, a3, a4) + a21det(e2, a2, a3, a4) + a31det(e3, a2, a3, a4) + a41det(e4, a2, a3, a4).

The first of the four determinants is

det(e1, a2, a3, a4) = det[ 1 a12 a13 a14 ] = det[ 1 0 0 0 ] = D(A11),
0 a22 a23 a24 0 a22 a23 a24
0 a32 a33 a34 0 a32 a33 a34
0 a42 a43 a44 0 a42 a43 a44

where we used the (just proved) column operations (-a12)[col 1] + [col 2], (-a13)[col 1] + [col 3], (-a14)[col 1] + [col 4] for the second equality, and

A11 = [ a22 a23 a24 ].
a32 a33 a34
a42 a43 a44

Since D(A11) is clearly multilinear and alternating in columns of A11, by this result, D(A11) = d detA11, where d = D(I3) = det(e1, e2, e3, e4) = detI4 = 1. Thus we conclude det(e1, a2, a3, a4) = D(A11) = detA11 = C11.

The other determinants can be computed by first using the (just proved) row operations to change to a determinant of the form det(e1, a2, a3, a4) and then use the computation above. For example,

det(e2, a2, a3, a4) = det[ 0 a12 a13 a14 ] = - det[ 1 a22 a23 a24 ] = - detA21 = C21.
1 a22 a23 a24 0 a12 a13 a14
0 a32 a33 a34 0 a32 a33 a34
0 a42 a43 a44 0 a42 a43 a44

At then end, we get

detA = a11C11 + a21C21 + a31C31 + a41C41.

For the expansions along some other columns, we may first use the (just proved) column operations to move the specific column to the first column and then use the result just proved. Finally, the idea of the argument for the 4 by 4 case clearly works for the general case.

Cofactor Expansion Along Row This follows from the column expansion case and the property detA = detAT proved before.

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