Question

If an n × n matrix A is similar to matrix B, then$A+7I_n$ must be similar to$\mathbf{B}+7I_n$.

Short answer

The above statement is true.

If an n × n matrix A is similar to matrix B, then $A+7I_n$must be similar to$B+7I_n.$

Step-by-step solution

Definition of similar matrix

Let A and B are two square matrices, the matrix A is said to be similar to matrix B if there exists an invertible matrix P such that

$B=P^{-1}AP$

Mentioning the concept

Since matrix A is similar to matrix B then there exists an invertible matrix P such that

$B=P^{-1}AP$

Now, we have to show that the matrix $A+7I_n$must be similar to the matrix$B+7I_n.$

We have

$$\begin{gathered} P^{-1}(A+7I_n)P=P^{-1}AP+P^{-1}7I_{n}P \\ \Rightarrow P^{-1}(A+7I_n)P=B+7I_n\text{(}\because P^{-1}AP=B\text{)} \end{gathered}$$

This implies that there exists an invertible matrix P such that

$P^{-1}(A+7I_n)P=B+I_n$

Hence, the matrix $A+7{]}_n$must be similar to the matrix$B+7I_n.$

Final Answer

If an n × n matrix A is similar to matrix B, then $A+7I_n$must be similar to$B+7I_n.$ as there exists an invertible matrix P such that$P^{-1}(A+7I_n)P=B+I_n.$