If A + B = $\left[\begin{array}{cc}1& 2\\ 5& -6\end{array}\right]$ and A − B = $\left[\begin{array}{cc}-3& 4\\ -1& -2\end{array}\right]$, then AB =

Concept Used:-

In the operation of addition of same order matrix, we add the corresponding elements of the matrices.

For examples, let's take two matrices,

$A=\left[\begin{array}{ll}{a}_1& a_2\\ a_3& a_4\end{array}\right]\text{ and }B=\left[\begin{array}{ll}{b}_1& b_2\\ b_3& b_4\end{array}\right]$

Then addition of this two matrix,

$$\begin{gathered} A+B=\left[\begin{array}{ll}{a}_1& a_2\\ a_3& a_4\end{array}\right]+\left[\begin{array}{ll}{b}_1& b_2\\ b_3& b_4\end{array}\right] \\ A+B=\left[\begin{array}{ll}{a}_1+b_1& a_2+b_2\\ a_3+b_3& a_4+b_4\end{array}\right] \end{gathered}$$

Similarly, we can find the subtraction of the matrix by subtracting the corresponding elements.

Explanation:-

Given that, 

A + B = $\left[\begin{array}{cc}1& 2\\ 5& -6\end{array}\right]$  ........(1) and

A − B = $\left[\begin{array}{cc}-3& 4\\ -1& -2\end{array}\right]$   ..............(2)

Adding both the equations,

⇒ A + B + A -B = $\left[\begin{array}{cc}1& 2\\ 5& -6\end{array}\right]$ + $\left[\begin{array}{cc}-3& 4\\ -1& -2\end{array}\right]$

⇒ 2A = $\left[\begin{array}{cc}1-3& 2+4\\ 5-1& -6-2\end{array}\right]$ 

⇒ 2A = $\left[\begin{array}{cc}-2& 6\\ 4& -8\end{array}\right]$

⇒ 2A = $2\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]$

Divide by 2 both sides we get

⇒ A =$\left[\begin{array}{cc}-1& 3\\ 2& -4\end{array}\right]$
Now subtract equation (2) from (1),

⇒ A + B - A + B = $\left[\begin{array}{cc}1& 2\\ 5& -6\end{array}\right]$ - $\left[\begin{array}{cc}-3& 4\\ -1& -2\end{array}\right]$

⇒ 2B = $\left[\begin{array}{cc}1+3& 2-4\\ 5+1& -6+2\end{array}\right]$ 

⇒ 2B = $\left[\begin{array}{cc}4& -2\\ 6& -4\end{array}\right]$

⇒ 2B = $2\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]$

Divide by 2 both sides we get

⇒ B =$\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]$

Now, the product of A and B,

⇒ AB =$\left[\begin{array}{cc}-1& 3\\ 2& -4\end{array}\right]$ $\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]$

⇒ AB =$\left(\begin{array}{ll}(-1)\cdot 2+3\cdot 3& (-1)(-1)+3(-2)\\ 2\cdot 2+(-4)\cdot 3& 2(-1)+(-4)(-2)\end{array}\right)$

⇒ AB = $\left(\begin{array}{cc}7& -5\\ -8& 6\end{array}\right)$

Hence, the correct option is 2.