16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.4.7

Determinants

Mathematics — Linear Algebra · block-diagonal-determinant

Problem

Let

A

be an

n \times n

matrix over

K

, a commutative ring with identity. Suppose

A

has the block form

A = \begin{bmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & & \vdots \\ 0 & 0 & \cdots & A_k \end{bmatrix}

where

A_i

is an

r_i \times r_i

matrix. Prove

\det A = (\det A_1)(\det A_2) \cdots (\det A_k).