Question

Question for Samaritans​

Answer 1

(A+B)^2 = A^2 + B^2 + 2AB not verified as AB not equal to BA

Step-by-step explanation:

A = 1 2

3 4

A^2 = 1*1 + 2*3 1*2 + 2*4

3*1 + 4*3 3*2 + 4*4

A^2 = 7 10

15 22

B^2 = 13 24

24 45

AB = 8 15

18 33

A^2 + 2AB + B^2

36. 64

75. 133

A+ B = 3 5

6 10

(A+B)^2 = 39 65

78 130

not equal

hence not verified

but if we find BA

BA = 11 16

21 30

then A^2 + B^2 + AB + BA

=39 65

78 130

hence

(A+B)^2 = A^2 + B^2 + AB + BA

if AB = BA

then only

(A+B)^2 = A^2 + B^2 + 2AB

Answer 2

✪ᴀɴsᴡᴇʀ✪

Here,

$A = [1 \:\: 2]$

$\:\:\:\:\:\:\:\:\: [3 \:\:4]$

$B = [2 \:\: 3]$

$\:\:\:\:\:\:\:\:\:\:[3 \:\:6]$

Now,

$(A+B)^2 = A^2 + 2AB + B^2$

if $A^2 + AB +BA+ B^2$

$\:\:\:\:\:= A^2 + 2AB + B^2$

if $AB +BA = 2AB$

if $BA = 2AB - AB$

if $BA = AB$

Thus it is sufficient to check if AB and BA are equal or not to verify the given equation.

So,

$AB = [1 \:\: 2] [2 \:\: 3]$

$\:\:\:\:\:\:\:\:\:\:\:\:\:[3 \:\:4] [3 \:\: 6]$

$AB = [1\times 2 + 2 \times 3\:\:\: 1\times 3 + 2 \times 6]$

$\:\:\:\:\:\:\:\:\:\:\:\:\:[3\times 2 + 4 \times 3\:\:\: 3\times 3 + 4 \times 6]$

$AB = [\:8 \:\:\: 15]$

$\:\:\:\:\:\:\:\:\:\:\:\:\:[18 \:\: 33]$

$BA = [2 \:\:3][1 \:\: 2]$

$\:\:\:\:\:\:\:\:\:\:\:\:\:[3 \:\: 6][3 \:\:4]$

$BA = [2\times 1 + 3 \times 3 \:\:\: 2\times 2 + 3 \times 4]$

$\:\:\:\:\:\:\:\:\:\:\:\:\:[3\times 1 + 6 \times 3\:\:\:3\times 2 + 6 \times 4]$

$BA = [11 \:\: 16]$

$\:\:\:\:\:\:\:\:\:\:\:\:\:[21 \:\:30]$

It is clear that $AB ≠ BA$. Hence it is verified that

$(A+B)^2 ≠ A^2 + 2AB + B^2$