Lower triangular matrix

A lower triangular matrix has zero entries above its main diagonal. Its determinant is the product of its diagonal entries, and an invertible lower triangular linear system is solved by forward substitution.

A lower triangular matrix with every diagonal entry equal to one. Its determinant is one, and its inverse is again unit lower triangular. Such matrices store the elimination multipliers in an LU decomposition.

For an invertible lower triangular matrix, the scalar pivot $a$ and the remaining triangular block $C$ are both invertible. The displayed formula follows by matrix multiplication: the lower-left block in the product is $b/a-C(C^{-1}b/a)=0$, and the diagonal blocks are identities. Recursively invert $C$, compute $-C^{-1}b/a$, and assemble the inverse matrix; the one-dimensional base case is $a^{-1}$. This recursion reduces the size by one at each step.

The algebra $T_n(k)$ consists of the lower triangular $n$ by $n$ matrices over $k$. Its Jacobson radical $N$ consists of the strictly lower triangular matrices, andThe $n$ simple left modules $S_r$ are one-dimensional, with a matrix acting through its $r$th diagonal entry. For the left regular module,