Question

If A and B are 4x4 matrices such that $A^{2}+B=A^{2}B$ then which of the following is correct?

• A. $AB=I$
• B. $A^{2}B=I$
• C. $A^{2}B=BA^{2}$
• D. $A^{2}=I$ or $B=I$

Hint:

If a matrix $B$ can be expressed as a polynomial or rational function of matrix $M$, then $B$ and $M$ will always commute.

Answer 1

The Correct Option is C

Step 1: Concept
Matrix commutation and algebraic manipulation of matrix equations.

Step 2: Meaning
The equation $A^{2} + B = A^{2}B$ can be rearranged to isolate the relationship between $A^{2}$ and $B$.

Step 3: Analysis
Rearrange the equation: $A^{2} = A^{2}B - B \implies A^{2} = (A^{2} - I)B$. Also, $B = A^{2}B - A^{2} \implies B = A^{2}(B - I)$. By manipulating these forms, we can test if $A^{2}$ and $B$ commute.

Step 4: Conclusion
From $A^{2}B - B = A^{2}$, we see $B$ is a function of $A^{2}$, specifically $B = (A^{2}-I)^{-1}A^{2}$ (assuming invertibility). Since any matrix commutes with its own inverse and power functions, $A^{2}B = BA^{2}$. Final Answer: (C)