Assertion (A) : If A = `({:(2,3),(1,4):}), B = ({:(1,0),(0,1):})`, then `(A+B)^(2) = A^(2) + B^(2) + 2 AB`.
Reason (R) : In the above AB = BA

Options

A. Both A and R are individually true and R is the correct explanation of A
B. Both A and R are individually true but R is not the correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

A

Explanation

`A=({:(2,3),(1,4):}),B=({:(1,0),(0,1):})`
`AB=({:(2,3),(1,4):})({:(1,0),(0,1):})`
`=({:(2+0,0+3),(1+0,0+4):})=({:(2,3),(1,4):})`
`BA=({:(1,0),(0,1):})({:(2,3),(1,4):})`
`=({:(2+0,3+0),(0+1,0+4):})=({:(2,3),(1,4):})`
Also,
`A+B=({:(3,3),(1,5):})`
`(A+B)^(2)=({:(9+3,9+15),(3+5,3+25):})({:(12,24),(8,28):})`
`A^(2)=({:(2,3),(1,4):})({:(2,3),(1,4):})=({:(4+3,6+12),(2+4,3+16):})=({:(7,18),(6,19):})`
`B^(2)=({:(1,0),(0,1):})({:(1,0),(0,1):})=({:(1,0),(0,1):})`
`A^(2)+B^(2)+2AB`
`=({:(7,18),(6,19):})+({:(1,0),(0,1):})+2({:(2,3),(1,4):})`
`=({:(12,24),(8,28):})=(A+B)^(2)`
So, Assertion A is correct
R is AB = BA
Hence, R is correct.
Since this leads from Assersion A, then both A and R are individually true and R is the correct explanation of A.