Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 1 / 2A (Vectors and Matrices)

The Leibniz formula for determinants isEquivalently, the determinant is the unique alternating function of the rows that is multilinear and takes value one on the identity matrix. Multiplying one row by $\lambda$ therefore gives

The scalar multiplication $\lambda A$ multiplies every one of the $n$ rows by $\lambda$. Applying multilinearity of the determinant once to each row gives

Interchanging two rows reverses the sign of an alternating multilinear map, soIn the Leibniz formula for determinants, the same result follows by composing every permutation with the corresponding transposition, which reverses its sign.

Let the columns be $C_1,\dots ,C_n$, and replace $C_k$ by $C_k+\lambda C_l$, where $k\ne l$. By multilinearity of the determinant,The second determinant vanishes because it has two equal columns. Hence