JAMIA 2026 — Mathematics PYQ

To solve this problem, we will use the property of idempotent matrices and the given matrix multiplication equations.

We are given two equations:

1. $AB=B$

2. $BA=A$

Step 1: Simplify $A^2$

Let's express $A^2$ using the given condition $A=BA$:

$$A^2=A\cdot A$$

Substitute $A=BA$ into the second part of the expression:

$$A^2=A\cdot (BA)$$

Using the associative property of matrix multiplication, we can regroup the terms:

$$A^2=(AB)\cdot A$$

Now, substitute the first given condition $AB=B$ into the expression:

$$A^2=B\cdot A$$

Since we know from the problem that $BA=A$, we get:

$$A^2=A$$

(Note: This means $A$ is an idempotent matrix.)

Step 2: Simplify $B^2$

Similarly, let's express $B^2$ using the given condition $B=AB$:

$$B^2=B\cdot B$$

Substitute $B=AB$ into the first part of the expression:

$$B^2=(AB)\cdot B$$

Using the associative property of matrix multiplication, regroup the terms:

$$B^2=A\cdot (BB)$$

Alternatively, substituting $B=AB$ into the second part gives:

$$B^2=B\cdot (AB)=(BA)\cdot B$$

Substitute the condition $BA=A$ into this equation:

$$B^2=A\cdot B$$

Since we know from the problem that $AB=B$, we get:

$$B^2=B$$

(Note: This means $B$ is also an idempotent matrix.)

Step 3: Calculate $A^2+B^2$

Now, add the simplified values of $A^2$ and $B^2$ together:

$$A^2+B^2=A+B$$

Correct Answer:

(b) A+B