Let A and B be 2 × 2 matrices. Then which of the following is true?

Calculation: 

Let 

and 

now 

det (A + B ) = (ad - bc) + (xw - yz) + aw + xd - bz - yc  . . ..(i) 

Now 

det (A - B) = ( ad - bc) + (xw - yz) - xd - aw + bz + yc   . . . .  (ii) 

Now from equations (i) and (ii), we get 

det (A + B ) + det(A - B) = 2 det(A) + 2 det (B) 

Hence option (3) is correct

Alternate MethodLet  and  

, 

then (1) ⇒ 1 + 1 = 1 + 0 (→ ←) - option (1) is false 

(4) ⇒ 1 - 1 = 2 - 0 (→ ←) option (4) is false

Let A = I_2×2, B = -I_2×2 then A + B = O_2×2, A - B = 2l_2×2

(2) ⇒ 0 + 2 = 2 - 2 (→ ←) option (2) is false.

So, option (3) is true.