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

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

2021-01-14
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
$$det(L^{-1}AL)=det(B)$$
$$det(L^{-1}AL)=det(B)$$
$$det(L^{-1})det(A)det(L)=det(B) \left[\text{ as det }(AB)=det(A)det(B)\right]$$
$$det(L^{-1})det(L)det(A)=det(B) \left[\text{ determinant of any matrix is a number so it is commutative }\right]$$
$$det(L^{-1}L)det(A)=det(B)$$
$$det(I)det(A)=det(B) \left[ det(I)=1\right]$$
$$det(A)=det(B)$$

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