Question

If A and B are n × n matrices, and vector$\overrightarrow{v}$ is in the image of both A and B, then$\overrightarrow{v}$ must be in the image of matrix A + B as well.

Short answer

The above statement is false.

If A and B are n × n matrices, and vector$\overrightarrow{v}$ is in the image of both A and B, then $\overrightarrow{v}$may not be in the image of matrix A + B.

Step-by-step solution

Given

We have given two n x n matrices A and B such that vector $\overrightarrow{v}$is in the image of both A and B.

Making Assumptions

Let us consider the matrix B = -A then the image of matrix A and matrix –A is the same.

Thus, if$\overrightarrow{v}$ is in the image of matrix A then$\overrightarrow{v}$ is also in the image of –A and hence,$\overrightarrow{v}$is in the image of both A and B = -A

Now, A + B = A + (-A) = 0

Thus, the image of (A+B) = 0, and hence vector$\overrightarrow{v}$is not in the image of A+B.

Final Answer

If we take B = -A such that vector$\overrightarrow{v}$is in the image of both A and B then A + B = 0 and hence the image of A + B = 0 yet A and B have the same image.

So, if A and B are n × n matrices, and vector$\overrightarrow{v}$ is in the image of both A and B, then $\overrightarrow{v}$may not be in the image of matrix A + B.