If A and B are n by n matrices, does (A+B)^2=A^2+2AB+B^2 in general?

Answer 1

No, because AB and BA are generally not equal. Matrix multiplication is not commutative.

So (A+B)^2 = A^2 + AB + BA + B^2 is the correct formula.

Answer 2

no,not in general,there is a condition:
if AB = BA => (A+B)^2 = (A+B)(A+B) = A^2 + AB + BA +B^2 =
A^2 + AB + AB + B^2 = A^2 + 2AB + B^2 = (A + B)^2

and:

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

Answer 3

If A and B are n by n matrices
then (A+B)^2 = (A+B)(A+B)
= A.A + A.B + B.A + B.B
= A^2+ A.B + B.A + B^2

and only if A.B = B.A then (A+B)^2=A^2+2AB+B^2 which is not always the case in generally.

so first one need to find whether A.B = B.A for above expression to be true.

If A=B then one can easily say above staement to be true.

also if both A and B are diagonal matrix then above statemnt holds....

Answer 4

No,

The correct answer would be A^2 + AB +BA + B^2

This is because matrices are not commutative that is

AB does not equal BA, Therefore you can not add them.

Answer 5

No, because in matrices AB is not equal to BA .
hence (A+B)^2 = (A+B)(A+B) = A.A + A.B + B.A + B.B
= A^2+ A.B + B.A + B^2

Answer 6

No. The square root of A+B would have to be a tangent to the cosine of 2AB.

Answer 7

Another two counter-examples are these one:

A (or B) = the nule matrix or the identity matrix

Ana

Answer 8

In general, no. However, in the case where A and B are permutable it is true.


Doug