Determinant of lower triangular matrix
Does a lower triangular matrix have a determinant that is equal to the product of the elements in the diagonal similar to an upper triangular matrix.
$\endgroup$ 21 Answer
$\begingroup$The matrix looks like this: $$\begin{bmatrix} a_{1,1}&0&\cdots&\cdots&0 \\b_{1,2}&a_{2,2}&\ddots&&\vdots \\ \vdots&\ddots&\ddots&\ddots&\vdots \\ \vdots&&\ddots&\ddots&0 \\ b_{1,n}&\cdots&\cdots&b_{n-1,n}&a_{n,n} \end{bmatrix}$$The determinant can be written as the sum of the product of the elements in the top row with their associated minors: so this determinant would be: $$a_{1,1}*\begin{bmatrix} a_{2,2}&0&\cdots&0 \\b_{2,3}&a_{3,3}&\ddots&\vdots \\ \vdots&\ddots&\ddots&0 \\ b_{2,n}&\cdots&b_{n-1,n}&a_{n,n} \end{bmatrix}$$This action eliminates the first column, $b_{1,x}$ and we are left with a similar matrix to find the determinant of. Reducing this one likewise we are left with $a_{1,1}*a_{2,2}*$(a matrix of remaining rows and columns) and so on until we have the product $a_{1,1}*a_{2,2}*\cdots*a_{n,n}$. The product of the diagonal.
$\endgroup$ 2
