# True or False - Determinants of two similar matrices are the same. Explain.

True or False - Determinants of two similar matrices are the same. Explain.
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SchulzD
Step 1
According to the given information, it is required to say whether the given statement is true or not.
The given statement is true as if two matrices are similar then their determinant are equal.
Step 2
Let A and B are similar matrices.
When two matrices are similar then there exists a non-singular matrix such that: let the non singular matrix be L then ,
${L}^{-1}AL=B$
take determinant both sides
$det\left({L}^{-1}AL\right)=det\left(B\right)$
$det\left({L}^{-1}AL\right)=det\left(B\right)$

$det\left({L}^{-1}L\right)det\left(A\right)=det\left(B\right)$
$det\left(I\right)det\left(A\right)=det\left(B\right)\left[det\left(I\right)=1\right]$
$det\left(A\right)=det\left(B\right)$
Jeffrey Jordon