- #1

- 120

- 26

- Homework Statement
- If A and B are two matrices such that AB=A and BA=B, then B^2 is equal to?

(a)A (b)B (c) 1 (d)0

- Relevant Equations
- A A^-1= I, where I is an identity matrix

I have a different way in solving the problem, but strangely, the result is different from that written in the solution manual.

My method:

Firstly, we will solve the ##AB=A## equation

$$AB=A$$

$$B=A^{−1}A$$

$$B=I$$

where ## I## is an identity matrix

Similarly, we can solve ##BA=B## using the same method

$$BA=B$$

$$A=B^{−1}B$$

$$A=I$$

where ## I ## is an identity matrix

It can be concluded that matrix ##A=B##. Hence, ##B^2=B=A##, and the answer to the multiple choice question is (a) and (b)

Now, let's look at the book's solution:

\begin{align}

B^2 & =BB\nonumber\\\

& =(BA)B\nonumber\\\

& = B(AB)\nonumber\\\

& =BA\nonumber\\\

&=B\nonumber

\end{align}

The answer is (b)

I thought that the matrix ##B## is equal to ##A##, but it seems that I am wrong(?)

Why both methods results in different answers?

My method:

Firstly, we will solve the ##AB=A## equation

$$AB=A$$

$$B=A^{−1}A$$

$$B=I$$

where ## I## is an identity matrix

Similarly, we can solve ##BA=B## using the same method

$$BA=B$$

$$A=B^{−1}B$$

$$A=I$$

where ## I ## is an identity matrix

It can be concluded that matrix ##A=B##. Hence, ##B^2=B=A##, and the answer to the multiple choice question is (a) and (b)

Now, let's look at the book's solution:

\begin{align}

B^2 & =BB\nonumber\\\

& =(BA)B\nonumber\\\

& = B(AB)\nonumber\\\

& =BA\nonumber\\\

&=B\nonumber

\end{align}

The answer is (b)

I thought that the matrix ##B## is equal to ##A##, but it seems that I am wrong(?)

Why both methods results in different answers?