Q.If $A = \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$, show that $(A + B)(A - B) \neq A^2 - B^2$.

Concept: Matrix multiplication is not commutative, so the identity $(A+B)(A-B)=A^2-B^2$ holds only if $A$ and $B$ commute ($AB=BA$). Here we verify they do not.

Step 1 – Compute $A+B$ and $A-B$:

$A+B = \begin{bmatrix}0&0\\2&1\end{bmatrix}$,

$A-B = \begin{bmatrix}0&2\\0&1\end{bmatrix}$.

Step 2 – Multiply $(A+B)(A-B)$:

$\begin{bmatrix}0&0\\2&1\end{bmatrix}\begin{bmatrix}0&2\\0&1\end{bmatrix} = \begin{bmatrix}0&0\\0&4+1\end{bmatrix} = \begin{bmatrix}0&0\\0&5\end{bmatrix}$.

Step 3 – Compute $A^2$ and $B^2$:

$A^2 = \begin{bmatrix}0&1\\1&1\end{bmatrix}\begin{bmatrix}0&1\\1&1\end{bmatrix} = \begin{bmatrix}1&1\\1&2\end{bmatrix}$,

$B^2 = \begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}0&-1\\1&0\end{bmatrix} = \begin{bmatrix}-1&0\\0&-1\end{bmatrix}$.

Step 4 – Compute $A^2 - B^2$: …

The core idea here is that the familiar algebraic identity $(a+b)(a-b)=a^2-b^2$ works for numbers because multiplication commutes — $ab = ba$. But matrices do not generally commute. So when you expand $(A+B)(A-B)$, you get:

The middle terms $-AB + BA$ cancel only if $AB = BA$. If they don't, the expression is different from $A^2 - B^2$. The problem asks us to verify this non-equality for the given matrices.

Let’s work through it step by step.

1. Compute $A+B$ and $A-B$

2. Compute $(A+B)(A-B)$

Multiply the two $2\times 2$ matrices:

• Row 1, Column 1: $(0)(0) + (0)(0) = 0$
• Row 1, Column 2: $(0)(2) + (0)(1) = 0$
• Row 2, Column 1: $(2)(0) + (1)(0) = 0$
• Row 2, Column 2: $(2)(2) + (1)(1) = 4 + 1 = 5$

So:

3. Compute $A^2$ and $B^2$ separately

First $A^2$:

Now $B^2$:

So $B^2 = -I$, which is a rotation by $90^\circ$ twice — indeed it gives the negative identity.

4. Compute $A^2 - B^2$ …

Method: Why $(A+B)(A-B)$ need not equal $A^2-B^2$

Use this reasoning for any product-identity that secretly assumes commutativity.

Steps

Step 1: Expand keeping the order of factors.

Step 2: Spot the commutativity assumption.

The middle terms $-AB+BA$ cancel only if $AB=BA$. For general matrices $AB\neq BA$, so the two sides differ. …

Common Mistakes

Mistake 1: Applying $a^2-b^2=(a+b)(a-b)$ to matrices.

Why it's wrong: the algebra identity assumes $ab=ba$; matrices generally do not commute, so $(A+B)(A-B)=A^2-AB+BA-B^2$. Correct approach: expand keeping the factor order.

Mistake 2: Cancelling $-AB+BA$ to zero.

Why it's wrong: this cancels only if $AB=BA$, which fails here. Correct approach: leave the middle terms and compute both sides explicitly. …