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We have any symmetric positive semi-definite matrices $A$ and $B$ where $\text{det}B=0$.

A function $f(t)=\text{det}(A+tB)$ is continuous on $t\in[0,1]$?

$f(t_{1})-f(t_{2})=\text{det}(A+t_{1}B)-\text{det}(A+t_{2}B)$. From here, I can't go farther. Thanks in advance.

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  • $\begingroup$ Do you know any formula for the determinant, or any calculation rule? $\endgroup$
    – Thomas
    Sep 9, 2018 at 7:21

1 Answer 1

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Expanding out $f(t)=\det(A+tB)$ using the formula for the determinant shows that $f(t)$ is a polynomial in $t$. All polynomials are continuous on all of $\Bbb R$. This is true for all matrices $A$ and $B$.

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  • $\begingroup$ Oh I see. Thanks! $\endgroup$
    – kayak
    Sep 9, 2018 at 7:22

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