- #1
ekkilop
- 29
- 0
The Cauchy expansion says that
[itex] \text{det} \begin{bmatrix}
A & x \\[0.3em]
y^T & a
\end{bmatrix}
= a \text{det}(A) - y^T \text{adj}(A) x [/itex],
where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar.
There is a proof in Matrix Analysis by Horn and Johnson that seems to be based on that A is a principal submatrix. My question is whether some similar result holds if A is not a principal submatrix? Say that we look for
det[itex] \begin{bmatrix}
y^T & a \\[0.3em]
A & x
\end{bmatrix}
[/itex].
Would a similar expression hold?
Thanks.
[itex] \text{det} \begin{bmatrix}
A & x \\[0.3em]
y^T & a
\end{bmatrix}
= a \text{det}(A) - y^T \text{adj}(A) x [/itex],
where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar.
There is a proof in Matrix Analysis by Horn and Johnson that seems to be based on that A is a principal submatrix. My question is whether some similar result holds if A is not a principal submatrix? Say that we look for
det[itex] \begin{bmatrix}
y^T & a \\[0.3em]
A & x
\end{bmatrix}
[/itex].
Would a similar expression hold?
Thanks.