Show that Gaussian elimination can be performed on A without row interchanges if and only if all leading principal sub matrices of A are nonsingular. [Hint: Partition each matrix in the equation

A^(k) = M^(k−1)M^(k−2) · · ·M⁽¹⁾A

Vertically between the kth and (k+1)st columns and horizontally between the kth and (k +1)st rows Show that the non singularity of the leading principal sub matrix of A is equivalent to a^(k)_k,k

≠0.]