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In-s [12.5K]
3 years ago
13

If A,B are n×n matrices, then (A + B)^2 = A^2 + 2AB + B^2. Either prove that this statement is true, or explain why it is false.

Mathematics
1 answer:
bonufazy [111]3 years ago
6 0
<h2>Answer:</h2>
  • The given statement is false.
  • The statement is necessarily true  when two matrix A and B commute.
<h2>Step-by-step explanation:</h2>

This statement is false.

Since, for two matrices A and B , the expansion of the term:

(A+B)^2 is given by:

(A+B)^2=(A+B)(A+B)\\\\i.e.\\\\(A+B)^2=A^2+B\cdot A+A\cdot B+B^2

Also, we know that when two matrix A and B commute then we have:

A\cdot B=B\cdot A

Hence, we get the expression as:

(A+B)^2=A^2+2A\cdot B+B^2

But when two matrix A and B do not commute then we need not have:

  (A+B)^2=A^2+2A\cdot B+B^2

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