Skip to main content
MathOverflow

Return to Question

TeX
Source Link
LSpice
LSpice
  • 12.9k
  • 4
  • 45
  • 69

Does linearity of cofactor imply linearity of determinant for 3x33×3 symmetric matrices?

If$\DeclareMathOperator\cof{cof}$If we suppose that a finite family (greater than 3) of 3X3$3\times3$ symmetric matrices A_i$A_i$ and positive reals a_i$a_i$ such that \sigma_ia_i=1$\sum_ia_i=1$ satisfies \cof(\sigma a_i A_i)=\sigma_ia_i\cof A_i$\cof(\sum_i a_i A_i)=\sum_ia_i\cof A_i$, does this imply that the same happenhappens for determinant? isIs it true that \det(\sigma a_i A_i)=\sigma_ia_i\det A_i$\det(\sum_i a_i A_i)=\sum_ia_i\det A_i$? ifIf not can we have, what is a counterexample. Thanks for reading!?

Does linearity of cofactor imply linearity of determinant for 3x3 symmetric matrices

If we suppose that a finite family (greater than 3) of 3X3 symmetric matrices A_i and positive reals a_i such that \sigma_ia_i=1 satisfies \cof(\sigma a_i A_i)=\sigma_ia_i\cof A_i does this imply that the same happen for determinant is it true that \det(\sigma a_i A_i)=\sigma_ia_i\det A_i if not can we have a counterexample. Thanks for reading!

Does linearity of cofactor imply linearity of determinant for 3×3 symmetric matrices?

$\DeclareMathOperator\cof{cof}$If we suppose that a finite family (greater than 3) of $3\times3$ symmetric matrices $A_i$ and positive reals $a_i$ such that $\sum_ia_i=1$ satisfies $\cof(\sum_i a_i A_i)=\sum_ia_i\cof A_i$, does this imply that the same happens for determinant? Is it true that $\det(\sum_i a_i A_i)=\sum_ia_i\det A_i$? If not, what is a counterexample?

Source Link
Yacine
Yacine
  • 19
  • 1

Does linearity of cofactor imply linearity of determinant for 3x3 symmetric matrices

If we suppose that a finite family (greater than 3) of 3X3 symmetric matrices A_i and positive reals a_i such that \sigma_ia_i=1 satisfies \cof(\sigma a_i A_i)=\sigma_ia_i\cof A_i does this imply that the same happen for determinant is it true that \det(\sigma a_i A_i)=\sigma_ia_i\det A_i if not can we have a counterexample. Thanks for reading!