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Option 1 should be correct. Since A$\neq$B and AB = BA hence A can have an inverse. Since if A had an inverse then

multiplying both sides of the equation AB = BA by A$^{-1}$ we get

$A^{-1}*AB = A^{-1} * BA$

$(A^{-1}*A)B = A^{-1} * AB$ // Since AB = BA

$I*B = (A^{-1} * A)B$

$B = I*B$

Which confirms the equality. Hence for sustenance of the equation A will have an inverse and hence

|A| $\neq$ 0

Now since A$^{2}$ = B$^{2}$

Hence

A$^{2}$ – B$^{2}$ = 0

A$^{2} + BA – BA - B^{2}$ = 0

A$^{2} + BA – AB - B^{2}$ = 0 // Since AB = BA

A$^{2} – AB + BA  - B^{2}$ = 0

A*(A – B) +B*(A – B) = 0

(A + B)*(A – B) = 0

Since A$\neq$B

A + B = 0

Hence |A+B| = 0

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