Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Determinant of a block triangular matrix

Builds on DeterminantBlock matrix

Statement

Let KK be a commutative ring, let AA be an m×mm \times m matrix, BB an m×nm \times n matrix and DD an n×nn \times n matrix over KK. Then the block matrix satisfies det⁡(AB0D)=det⁡A⋅det⁡D,\det \begin{pmatrix} A & B \\ 0 & D \end{pmatrix} = \det A \cdot \det D, where 00 is the n×mn \times m zero matrix and det⁡\det is the determinant.

Remarks

The same holds for the lower block triangular matrix (A0CD)\begin{pmatrix} A & 0 \\ C & D \end{pmatrix}. It generalizes the rule for triangular matrices. For a general block matrix, det⁡(ABCD)≠det⁡Adet⁡D−det⁡Bdet⁡C\det \begin{pmatrix} A & B \\ C & D \end{pmatrix} \ne \det A \det D - \det B \det C in general.

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  1. Identity matrix object
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  10. Determinant of a block triangular matrix statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Block_matrix, Determinant
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Tag0J9TVTNB4K
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Kindstatement, proposition
WritingW0 stub
MSC15A15
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