Properties of determinants

A determinant with a row or column of zeroes is 0.

Proof: Any elementary product must contain a number from that row or column.

A determinant with two equal rows or two equal columns is 0.

Proof: Replace one of the rows or columns by itself minus the other to get a row or column of zeroes.

A determinant with two proportional rows or two proportional columns is 0.

Proof: Factor out the proportionality constant to get a determinant with two equal rows.

If A is an n x n matrix and c a scalar, then det(cA) = cndet(A).

Proof: Factor the scalar c out of each of the n rows.

If A is an n x n matrix, then det(–A) = (–1)ndet(A).

Proof: Factor a (–1) out of each of the n rows.

The most important and useful determinant property of all.

A matrix has an inverse if and only if its determinant is not zero.

Proof: Key point: row operations don't change whether or not a determinant is 0; at most they change the determinant by a non-zero factor or change its sign.

Use row operations to reduce the matrix to reduced row-echelon form.

If the matrix is invertible, you get the identity matix, with non-zero determinant 1, so the original matrix had a non-zero determinant.

If the matrix is not invertible, you get a reduced form with a row of zeroes, which has determinant 0, so the original matrix had determinant 0.

Another very important determinant property and a consequence.

The determinant of the product of two matrices is the product of their determinants:
det(AB) = det(A)det(B).

Proof: Omitted. (Uses elementary matrices.)

If A is invertible, then det(A–1) = 1/det(A).

Proof: This follows from the first part and the relation A–1A = I.