Suppose A is a block upper triangular matrix Solution A is a upper triangular matrix If we expand by I column, except a11 all other elements in column I is 0 Hence Det A = a11 * cofactor of a11 Consider cofactor of a11. It will also be upper triangular, with top entry as a22. Expand the cofactor matrix by I column. a22* cofactor of a22 This procedure goes like a chain and hence det A = a11(a22)(a33)...(ann) = Product of all diagonal elements ------------------------------------------------------------------------------ det A1 = block hence along diagonal square matrix det A1 = a11*a22*..akk det A2 = ak+1*ak+2*... and so on Finally det A1*det A2*..... = product of all diagonal elements of A = det A Hence proved.

1. Suppose A is a block upper triangular matrix
Solution
A is a upper triangular matrix
If we expand by I column, except a11 all other elements in column I is 0
Hence Det A = a11 * cofactor of a11
Consider cofactor of a11. It will also be upper triangular, with top entry as a22.
Expand the cofactor matrix by I column.
a22* cofactor of a22
This procedure goes like a chain and hence
det A = a11(a22)(a33)...(ann) = Product of all diagonal elements
------------------------------------------------------------------------------
det A1 = block hence along diagonal square matrix
det A1 = a11*a22*..akk
det A2 = ak+1*ak+2*... and so on
Finally det A1*det A2*..... = product of all diagonal elements of A
= det A
Hence proved