Determinant and Inverse of a block matrix with diagonal blocks

Question

So I want to calculate the inverse and determinant of a block matrix ($n\times n$) with diagonal matrices (of same size) as block. Although I have a general idea as to how to calculate the determinant. I can't find anything relevant to find the inverse of such a matrix.

I plan to calculate the determinant by first rearranging the columns, rows to make this matrix a diagonal block matrix, then will calculate the determinant of each of the blocks and multiply them.

Any help in how to calculate the inverse as well as whether my approach to calculating determinant is correct or not is greatly appreciated.

TIA.

Answer 1

The direct sum of $n$ copies of the field of scalars are represented by $n\times n$ diagonal matrices. These matrices form a commutative ring with unit denoted by $R$. The matrices

block matrix ($n\times n$) with diagonal matrices (of same size) as block

you describe are matrices can also be interpreted as matrices with elements in $R$. The usual matrix operations of addition, multiplication and division give the same results if the matrices are regarded as matrices with elements in $R$.

You asked

Any help in how to calculate the inverse as well as whether my approach to calculating determinant is correct or not is greatly appreciated.

A similar result holds for determinants but in the final step you take the determinant of the resulting matrix in $R$ which is the product of all of the diagonal elements.