Let A and B be square matrices of the same order. Does (A + B)2 = A2 + 2AB + B2 hold? If not, why?
Given that A and B are square matrices of the same order.
We need to check if (A + B)2 = A2 + 2AB + B2.
We know (A + B)2 = (A + B)(A + B)
⇒ (A + B)2 = A(A + B) + B(A + B)
∴ (A + B)2 = A2 + AB + BA + B2
For the equation (A + B)2 = A2 + 2AB + B2 to hold, we need AB = BA that is the matrices A and B must satisfy the commutative property for multiplication.
However, here it is not mentioned that AB = BA.
Therefore, AB ≠ BA.
Thus, (A + B)2 ≠ A2 + 2AB + B2.