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The below statement is a true/false exercise.

Statement: For all square matrices A and B of the same size, it is true that $(A + B)2 = A^2 + 2AB + B^2$.

My thought process: Since it is not a proof, I figure I can show by example and come to a valid conclusion based on such example.

My work:

Come up with a square matrix A and B let both be a 2 by 2 matrix(rows and cols must be same).

Matrix $A$:

$A = \begin{array}{ccc} 3 & 5 \\ 4 & 6 \\ \end{array} $

Matrix $B$:

$B = \begin{array}{ccc} 5 & 8 \\ 9 & 4 \\ \end{array} $

$A + B = \begin{array}{ccc} 8 & 13 \\ 13 & 10 \\ \end{array}$

$(A + B)^2 = \begin{array}{ccc} 233 & 234 \\ 234 & 264 \\ \end{array}$

$A^2 = \begin{array}{ccc} 29 & 45 \\ 36 & 56 \\ \end{array}$

$(AB) = \begin{array}{ccc} 60 & 44 \\ 74 & 56 \\ \end{array}$

$2(AB) = \begin{array}{ccc} 120 & 88 \\ 234 & 112 \\ \end{array}$

$B^2 = \begin{array}{ccc} 97 & 72 \\ 81 & 88 \\ \end{array}$

$A^2 + 2AB + B^2 = \begin{array}{ccc} 246 & 205 \\ 265 & 256 \\ \end{array}$

Based my above work, the answer is false.

Is there another way to approach the problem? It seems like a lot of work needed to be done for a true/false question which raised my suspicion about whether there is a better way to look at the problem.

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    $\begingroup$ First, since a counterexample is all what you need, you can choose simpler matrices. With a lot of $0$, for instance. $\endgroup$
    – alex
    Mar 4, 2014 at 22:01
  • $\begingroup$ It is well known that matrix multiplication is not commutative. $\endgroup$
    – EricAm
    Mar 4, 2014 at 22:01
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    $\begingroup$ Second, ask yourself: where does the rule $(a+b)^2=a^2+2ab+b^2$ about real numbers come from? $\endgroup$
    – alex
    Mar 4, 2014 at 22:03
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    $\begingroup$ @alex distrubutive law for addition. But the problem gives it in terms of matrices. I did not even realized there was a rule hidden in the problem until you ask the question. $\endgroup$
    – Nicholas
    Mar 4, 2014 at 22:15

2 Answers 2

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A counterexample is all you need, so you're done. You probably could have picked a simpler counterexample, say with most entries of $A$ and $B$ being $0,$ but yours works just fine.

As an alternative, note that if $A$ and $B$ are square matrices of the same size, then $$(A+B)^2=A(A+B)+B(A+B)=A^2+AB+BA+B^2.$$ From this, it follows that $$(A+B)^2=A^2+2AB+B^2$$ if and only if $AB=BA.$ So, any two square matrices $A,B$ of the same size such that $AB\neq BA$ will yield a counterexample.

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  • $\begingroup$ Holy cow! I never thought of it like that. By the way, how did you know right off the bat the distrubitive law of addition will work for square matrices if AB = BA? $\endgroup$
    – Nicholas
    Mar 4, 2014 at 22:33
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    $\begingroup$ @Nicholas because distributiveness works even when $AB\ne BA$ (it's not related) $\endgroup$ Mar 4, 2014 at 22:51
  • $\begingroup$ @ratchetfreak Oh okay. I thought there was a relationship to be drawn. $\endgroup$
    – Nicholas
    Mar 4, 2014 at 23:05
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    $\begingroup$ @Nicholas: It's worth noting that this problem demonstrates that commutativity of multiplication is necessary to conclude that $(a+b)^2=a^2+2ab+b^2,$ where $a,b\in\Bbb R.$ Distributivity isn't enough, as we see above. $\endgroup$ Mar 5, 2014 at 4:50
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    $\begingroup$ @Ahmed: Yes, associativity of addition is required. $\endgroup$ Sep 4, 2016 at 0:45
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$$\begin{align*} (A+B)^2 &= (A+B)(A+B) \\ &= AA+AB+BA+BB \\ &= A^2 + AB+BA+B^2.\end{align*}$$

Is it always true that $AB+BA = 2AB$?

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    $\begingroup$ Of course, you'd probably show the question false by using a counterexample... $\endgroup$
    – apnorton
    Mar 4, 2014 at 22:10
  • $\begingroup$ @Arkamis Based on my counterexample, it does not look to be always be true. $\endgroup$
    – Nicholas
    Mar 4, 2014 at 22:23

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