I need help with this excersice: "Suppose $L$ and $M$ are lower triangular matrices with ones on the diagonal. Show that $LM$ is lower triangular with ones on the diagonal" This exercise is taken from Elementary Linear Algebra - Stanley I. Grossman 7th edition.
[Hint: If $B=LM$ prove that $b_{ii}=\sum\limits_{k=1}^{n}l_{ik}m_{ki}=1$ and $b_{ij}=\sum\limits_{k=1}^{n}l_{ik}m_{kj}=0$ , if $j>i$].
But I'm stuck, I really don't know where to start. I really apreciate for your help.
Let us call a lower triangular matrix with a diagonal of 1 a LD1-matrix.
Instead of working with indices, I advise you to prove it by induction on the size $n$ of the matrices considered as an upper left block of size $n-1$ bordered by a line vector, a column vector and a real entry, like this:
$$\underbrace{\begin{pmatrix}L_1&0\\U&1\end{pmatrix}}_L\underbrace{\begin{pmatrix}L_2&0\\V&1\end{pmatrix}}_M=\underbrace{\begin{pmatrix}L_3&S\\W&a\end{pmatrix}}_B\tag{1}$$
with the aim to prove three
Objectives: prove that $L_3$ is an LD1-matrix, $S=0$ (null vector) and $a=1.$
(in this way $B$ will be a LD1-matrix of order $n$).
Let us write down the 4 block equations equivalent to (1),
$$\begin{cases}L_1L_2&=&L_3\\0&=&S\\UL_2+V&=&W\\1&=&a\end{cases}\tag{2}$$
and make a reasoning by induction (i.e., we assume that $L_1,L_2$ are LD1-matrices.).
The first equation in (2) is where the induction step is used: $L_3$ is an LD1 matrix as product of two LD1 matrices (with dimension $n-1$). The second and fourth equations in (2) fullfill the two other objectives.
Remark: The third equation doesn't bring information.