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Kunze Linear Algebra
· Problem 5.4.7
Determinants
Mathematics — Methods · block-diagonal-determinant
Problem
Let
A
A
A
be an
n
×
n
n \times n
n
×
n
matrix over
K
K
K
, a commutative ring with identity. Suppose
A
A
A
has the block form
A
=
[
A
1
0
⋯
0
0
A
2
⋯
0
⋮
⋮
⋮
0
0
⋯
A
k
]
A = \begin{bmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & & \vdots \\ 0 & 0 & \cdots & A_k \end{bmatrix}
A
=
A
1
0
⋮
0
0
A
2
⋮
0
⋯
⋯
⋯
0
0
⋮
A
k
where
A
i
A_i
A
i
is an
r
i
×
r
i
r_i \times r_i
r
i
×
r
i
matrix. Prove
det
A
=
(
det
A
1
)
(
det
A
2
)
⋯
(
det
A
k
)
.
\det A = (\det A_1)(\det A_2) \cdots (\det A_k).
det
A
=
(
det
A
1
)
(
det
A
2
)
⋯
(
det
A
k
)
.
Related theory
Diagonal Relationships In The P Block
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