# Show that if A and B are similar, then det A=det B.

Alyce Wilkinson 2021-11-05 Answered
Show that if A and B are similar, then det A=det B.

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hajavaF
Consider two similar matrix P and Q
If two matrix are similar, then there wxist a singular matrix 'A'.
$$\displaystyle{Q}={A}^{{-{1}}}{P}{A}$$
If P and Q are $$\displaystyle{m}\times{m}$$ matrix, then Det(PQ)=Det P * Det Q
$$\displaystyle{D}{e}{t}{P}={D}{e}{t}{\left({A}^{{-{1}}}{Q}{A}\right)}$$
$$\displaystyle={\left({D}{e}{t}{A}^{{-{1}}}\right)}{\left({D}{e}{t}{Q}\right)}{\left({D}{e}{t}{A}\right)}$$
$$\displaystyle={\left({D}{e}{t}{Q}\right)}{\left({D}{e}{t}{A}^{{-{1}}}\right)}{\left({D}{e}{t}{A}\right)}$$
$$\displaystyle={\left({D}{e}{t}{Q}\right)}{\left({D}{e}{t}\forall^{{-{1}}}\right)}$$
=(Det Q)(Det I)
Det P=Det Q
Results:
Determinant of similar matrices is equal