Suppose that $A$ and $B$ are similar. Then there exists a nonsingular matrix $S$ such that
\[S^{-1}AS=B\]
by definition.
Then we have
\begin{align*}
&\det(B)\\
&=\det(S^{-1}AS)\\
&=\det(S)^{-1}\det(A)\det(S) \\
& \text{(by multiplicative properties of determinants)}\\
&=\det(A) \\
&\text{(since determinants are just numbers, hence commutative).}
\end{align*}

Thus, we obtain $\det(A)=\det(B)$ as required.

Related Question.

More generally, we can prove that if $A$ and $B$ are similar, then their characteristic polynomials are the same.
From this, we also can deduce that the determinants of $A$ and $B$ are the same as well as their traces are the same.

Determine Whether Given Matrices are Similar
(a) Is the matrix $A=\begin{bmatrix}
1 & 2\\
0& 3
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
3 & 0\\
1& 2
\end{bmatrix}$?
(b) Is the matrix $A=\begin{bmatrix}
0 & 1\\
5& 3
\end{bmatrix}$ similar to the matrix […]

If 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero Matrix
Let $A, B$ be complex $2\times 2$ matrices satisfying the relation
\[A=AB-BA.\]
Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.
Hint.
Find the trace of $A$.
Use the Cayley-Hamilton theorem
Proof.
We first calculate the […]

True or False: If $A, B$ are 2 by 2 Matrices such that $(AB)^2=O$, then $(BA)^2=O$
Let $A$ and $B$ be $2\times 2$ matrices such that $(AB)^2=O$, where $O$ is the $2\times 2$ zero matrix.
Determine whether $(BA)^2$ must be $O$ as well. If so, prove it. If not, give a counter example.
Proof.
It is true that the matrix $(BA)^2$ must be the zero […]

Trace, Determinant, and Eigenvalue (Harvard University Exam Problem)
(a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$.
Find $\det(A)$.
(b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.
(c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of […]

A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable
Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.
Definitions/Hint.
Recall the relevant definitions.
Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]

Determinant/Trace and Eigenvalues of a Matrix
Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.
Show that
(1) $$\det(A)=\prod_{i=1}^n \lambda_i$$
(2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$
Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix […]

Matrix $XY-YX$ Never Be the Identity Matrix
Let $I$ be the $n\times n$ identity matrix, where $n$ is a positive integer. Prove that there are no $n\times n$ matrices $X$ and $Y$ such that
\[XY-YX=I.\]
Hint.
Suppose that such matrices exist and consider the trace of the matrix $XY-YX$.
Recall that the trace of […]

Dimension of Null Spaces of Similar Matrices are the Same
Suppose that $n\times n$ matrices $A$ and $B$ are similar.
Then show that the nullity of $A$ is equal to the nullity of $B$.
In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of […]

[…] that if $A$ and $B$ are similar, then their determinants are the same. We compute begin{align*} det(A)=(1)(3)-(2)(0)=3 text{ and } det(B)=(3)(2)-(0)(1)=6. […]

## 1 Response

[…] that if $A$ and $B$ are similar, then their determinants are the same. We compute begin{align*} det(A)=(1)(3)-(2)(0)=3 text{ and } det(B)=(3)(2)-(0)(1)=6. […]