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

Let $A,B,C$ be matrices of the same order (conformable for the operations below) and $a,b$ scalars.

Addition/scalar multiplication:

1. $A+B$ is again a matrix of the same order.
2. $A+B=B+A$ (commutative).
3. $(A+B)+C=A+(B+C)$ (associative).
4. $A+O=O+A=A$ ($O$ is the additive identity).
5. $A+(-A)=O=(-A)+A$ ($-A$ is the additive inverse).
6. $(a+b)A = aA+bA$ and $a(A+B)=aA+aB$.
7. $a(bA)=(ab)A$, $1A=A$, and $0A=O$.

Multiplication (when the orders are conformable):

• Associative: if $A$ is $m\times n$, $B$ is $n\times p$, $C$ is $p\times q$, then $A(BC)=(AB)C$, a matrix of order $m\times q$.
• Left distributive: $A(B+C)=AB+AC$.
• Right distributive: $(A+B)C=AC+BC$.
• For a scalar $\alpha$: $\alpha(AB) = (\alpha A)B = A(\alpha B)$.
• $AI=IA=A$ where $I$ is the unit matrix ($I$ is the multiplicative identity). …