Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 12 / viii / Solution

Let $M$ be the square matrix whose $j$th column is $A_j$. The Leibniz formula for determinants is precisely the stated contraction:A square matrix has zero determinant exactly when its columns are linearly dependent, proving the required equivalence. One direction also follows directly from multilinearity and antisymmetry of a multilinear map: replacing one column by a linear combination of the others leaves only terms with repeated columns, each zero. Conversely, nonzero determinant gives an inverse matrix and hence independent columns.