Question

inverse of a lower-triangular matrix is lower triangular proof

Answer 1

L−1=[y1⋯yn],

where each yk is an n×1 matrix.

Now, by definition,

LL−1=I=[e1⋯en],

where ek is the n×1 matrix with a 1 in the kth row and 0s everywhere else. Observe, though, that

LL−1=L[y1⋯yn]=[Ly1⋯Lyn],

so

Lyk=ek(1≤k≤n)

By the proposition, since ek has only 0s above the kth row and L is lower triangular and Lyk=ek, then yk has only 0s above the kth row. This is true for all 1≤k≤n, so since

L−1=[y1⋯yn],

then L−1 is lower triangular