CBSE 2025, Set 3

Question

If $A$ and $B$ are square matrices of same order such that $AB=A$ and $BA=B$, then $A^{2}+B^{2}$ is equal to : A). $A+B$ B). $BA$ C). $2(A+B)$ D). $2BA$

Edvaya · Accepted Answer

The correct answer is $A+B$. **Solution:**

Given the matrix equations: 
1) $AB = A$
2) $BA = B$

We need to find the value of $A^2 + B^2$.

**Step 1: Simplify $A^2$**
$$A^2 = A \cdot A$$
Substitute the first $A$ with $AB$ (from equation 1):
$$A^2 = (AB)A$$
By associativity of matrix multiplication:
$$A^2 = A(BA)$$
Substitute $BA$ with $B$ (from equation 2):
$$A^2 = AB$$
Using equation 1 again:
$$A^2 = A$$

**Step 2: Simplify $B^2$**
$$B^2 = B \cdot B$$
Substitute the first $B$ with $BA$ (from equation 2):
$$B^2 = (BA)B$$
By associativity:
$$B^2 = B(AB)$$
Substitute $AB$ with $A$ (from equation 1):
$$B^2 = BA$$
Using equation 2 again:
$$B^2 = B$$

**Step 3: Add the terms**
$$A^2 + B^2 = A + B$$

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