True. To see this, just write out the matrices in terms of their components, and do the math. It's a bit tedious, but since they're just 2x2 it's not too bad:
AC+BC
=
[a1 a2][c1 c2] + [b1 b2][c1 c2]
[a3 a4][c3 c4] + [b3 b4][c3 c4]
=
[a1c1 + a2c3 a1c2 + a2c4] + [b1c1 + b2c3 b1c2 + b2c4]
[a3c1 + a4c3 a3c2 + a4c4] + [b3c1 + b4c3 b3c2 + b4c4]
=
[(a1 + b1)c1 + (a2 + b2)c3 (a1 + b1)c2 + (a2 + b2)c4]
[(a3 + b3)c1 + (a4 + b4)c3 (a3 + b3)c2 + (a4 + b4)c4]
=
[a1+b1 a2+b2][c1 c2]
[a3+b3 a4+b4][c3 c4]
=
(A+B)CGiven matrices A, B, and C are all 2x2, determine whether the equation is true. AC+BC=(A+B)C True or False?
That one turns out to be true, and it remains true no matter how big A, B, and C are (so long as the sums and products on both sides of the equation are defined). Until you get used to it, it can be a little difficult to tell which identities work and which ones don't, but in my opinion the best way is to try to think about the matrices as linear transformations (if you are comfortable with that) and then manipulate them as you would functions. Remember that matrix multiplication corresponds to composition of linear transformations, and all kinds of weird things can happen when you compose two functions.Given matrices A, B, and C are all 2x2, determine whether the equation is true. AC+BC=(A+B)C True or False?
False
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