Q.Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify $4AB + 3(AB + BA) - 4BA$, where A and B are both matrices of order $2 \times 2$. It is known that $A \neq B \neq I$ and $A^{-1} \neq B$. Their answers are given as. Who answered it correctly? (A) Abhay: $6AB$ (B) Bina: $7AB - BA$ (C) Chhaya: $8AB$ (D) Devesh: $7BA - AB$

The key idea is the distributive property of matrix multiplication and combining like terms, recognizing that $AB$ and $BA$ are generally distinct matrix products.

1. Distribute the scalar $3$ into the parenthesis: $4AB + 3(AB + BA) - 4BA = 4AB + 3AB + 3BA - 4BA$
2. Group the terms involving $AB$ and the terms involving $BA$: $(4AB + 3AB) + (3BA - 4BA)$
3. Combine the coefficients for each type of term: $(4+3)AB + (3-4)BA = 7AB - 1BA$ …

When working with matrices, it is crucial to remember that matrix multiplication is generally not commutative. This means that for two matrices $A$ and $B$, the product $AB$ is usually not equal to $BA$. While there are specific cases where $AB = BA$ (for example, if $A$ is the identity matrix or if $A$ and $B$ are inverse matrices, or if they simply happen to commute), we cannot assume this property unless it is explicitly stated or can be derived.

In this problem, we are given matrices $A$ and $B$ of order $2 \times 2$, and it is stated that $A \neq B \neq I$ and $A^{-1} \neq B$. These conditions reinforce that we should not assume $AB = BA$. Therefore, $AB$ and $BA$ must be treated as distinct terms, similar to how $x$ and $y$ are distinct terms in an algebraic expression. We can only combine terms that are exactly alike.

Let's simplify the given expression step-by-step:

1. Write down the given expression:

   The expression to simplify is $4AB + 3(AB + BA) - 4BA$.

2. Distribute the scalar:

   First, distribute the scalar $3$ into the parenthesis $(AB + BA)$:

   $4AB + (3 \times AB) + (3 \times BA) - 4BA$

   This simplifies to:

   $4AB + 3AB + 3BA - 4BA$

3. Group like terms:

   Now, group the terms involving $AB$ together and the terms involving $BA$ together. Remember, $AB$ and $BA$ are treated as distinct entities.

   $(4AB + 3AB) + (3BA - 4BA)$

4. Combine coefficients of like terms: …