Q.If $A=\begin{bmatrix}1&2\\-1&-2\end{bmatrix}$, $B=\begin{bmatrix}2&a\\-1&b\end{bmatrix}$ and if $(A+B)^2=A^2+B^2$. find values of a and b.

Given $A=\begin{bmatrix}1&2\\-1&-2\end{bmatrix}$, $B=\begin{bmatrix}2&a\\-1&b\end{bmatrix}$.

In general $(A+B)^2=A^2+AB+BA+B^2$. For this to equal $A^2+B^2$, we need $AB+BA=O$.

Compute AB:

$(AB)_{11}=1(2)+2(-1)=2-2=0$

$(AB)_{12}=1(a)+2(b)=a+2b$

$(AB)_{21}=-1(2)+(-2)(-1)=-2+2=0$

$(AB)_{22}=-1(a)+(-2)(b)=-a-2b$

$AB=\begin{bmatrix}0&a+2b\\0&-a-2b\end{bmatrix}$

Compute BA:

$(BA)_{11}=2(1)+a(-1)=2-a$

$(BA)_{12}=2(2)+a(-2)=4-2a$

$(BA)_{21}=-1(1)+b(-1)=-1-b$

$(BA)_{22}=-1(2)+b(-2)=-2-2b$

$BA=\begin{bmatrix}2-a&4-2a\\-1-b&-2-2b\end{bmatrix}$

Add and set to O:

$AB+BA=\begin{bmatrix}2-a&-a+2b+4\\-1-b&-a-4b-2\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$

From $(1,1)$: $2-a=0\Rightarrow a=2$

From $(2,1)$: $-1-b=0\Rightarrow b=-1$

Check $(1,2)$: $-a+2b+4=-2-2+4=0$ ✓

Check $(2,2)$: $-a-4b-2=-2+4-2=0$ ✓ …

Expand (A+B)² symbolically; since (A+B)²=A²+AB+BA+B², the given equation forces AB+BA=O, so set that sum …

• Expanding $(A+B)^2$ as $A^2+2AB+B^2$ (valid for numbers, not matrices) instead of the correct $A^2+AB+BA+B^2$. …