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IF $A$ and $B$ are two matrices, find $(A-B)^2$.

My first solution was : $$ A^2 - 2AB + B^2. $$

My second solution was : $$ (A-B)(A-B) \\ A^2 - AB - BA + A^2 $$

Is one of them correct ? or both of them are incorrect ?

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    $\begingroup$ Are you trying to find $(A + B)^2$ or $(A - B)^2$? You need to edit your post to make it clear. As it stands, both your answers are wrong if you are trying to find $(A + B)^2$ $\endgroup$
    – user137481
    May 11, 2014 at 19:12

3 Answers 3

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$A^2 - 2AB + B^2$ is not true for all matrices.

$A^2 - AB - BA + B^2$ is correct.

You need to know that $AB = BA$ in order to conclude that $(A-B)^2 = A^2 - 2AB + B^2$.

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$(A-B)^2=(A-B)(A-B)=A(A-B)-B(A-B)=A^2-AB-BA+B^2$. In general $AB\not=BA$.

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In general, the second solution is correct (with $B^2$ instead of $A^2$ at the end) but the first one is incorrect because matrix multiplication isn't commutative.

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  • $\begingroup$ I meant that .. Thanks $\endgroup$
    – Maher
    May 11, 2014 at 19:12

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