Question

There exists a non-zero 2 x 2 matrix A that is similar to 2A.

Short answer

The above statement is true.

There exists a non-zero 2 x 2 matrix A that is similar to 2A.

Step-by-step solution

Definition of similar matrix

Let us suppose two square matrices A and B. Then the matrix B is said to be similar to the matrix A if there exists an invertible matrix P such that

$$\begin{gathered} B=P^{-1}AP \\ A=P^{-1}\left(2A\right)P \end{gathered}$$

Finding an invertible matrix

Let us suppose,

$A=\left[\begin{array}{cc}0& 2\\ 0& 0\end{array}\right]\text{and}2A=\left[\begin{array}{cc}0& 4\\ 0& 0\end{array}\right]$

Also, let the matrix Abesimilar matrix to the matrix 2A then there exists an invertible matrix $P=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$such that

AP=P(2A)

$$\begin{gathered} \Rightarrow \left[\begin{array}{cc}0& 2\\ 0& 0\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}0& 4\\ 0& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}2c& 2d\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 4a\\ 0& 4c\end{array}\right] \\ \Rightarrow 2c=0\Rightarrow c=0 \\ 2d=4a\Rightarrow d=2a \end{gathered}$$

Thus, the matrix $P=\left[\begin{array}{cc}a& b\\ 0& 2a\end{array}\right]$is an invertible matrix for$a\ne 0$ .

There existsaninvertible matrix $P=\left[\begin{array}{cc}a& b\\ 0& 2a\end{array}\right]$for$a\ne 0$ such that

$A=P^{-1}\left(2A\right)P$

Final Answer

If $A=\left[\begin{array}{cc}0& 2\\ 0& 0\end{array}\right]\text{and}2A=\left[\begin{array}{cc}0& 4\\ 0& 0\end{array}\right]$then there exists an invertible matrix$P=\left[\begin{array}{cc}a& b\\ 0& 2a\end{array}\right]$ for such that

$A=P^{-1}\left(2A\right)P$

Hence, matrix A is similar to matrix 2A.

So, there exists a non-zero 2 x 2 matrix A that is similar to 2A.