Question

If $A$ and $B$ square matrices of order 2, then ${(A + B)^2} = $
A. ${A^2} - 2AB + {B^2}$
B. ${A^2} + 2AB + {B^2}$
C. ${A^2} + 2BA + {B^2}$
D. None of these

Answer 1

Hint: In this question, we have to find out the value of ${(A + B)^2}$. To determine that, we must know the basic properties of the matrix and algebraic identities. To find the answer to the given problem, we will expand the expression given in the question.

Complete step by step Solution:
Given that \[A\] and \[B\] are square matrices.
Now, on expanding the given expression using algebraic identity, we get:
${(A + B)^2} = {A^2} + 2AB + {B^2} \ldots \ldots \ldots \ldots eq(1)$ $[\because {(a + b)^2} = {a^2} + 2ab + {b^2}]$

As we know, in algebra, $ab = ba$, but, in matrix, $AB \ne BA$
For example, let’s consider two matrices, matrix $A = \left[ {\begin{array}{*{20}{c}}
  1&2 \\
  3&4
\end{array}} \right]$ and matrix $B = \left[ {\begin{array}{*{20}{c}}
  5&6 \\
  7&8
\end{array}} \right]$
$AB = \left[ {\begin{array}{*{20}{c}}
  1&2 \\
  3&4
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
  5&6 \\
  7&8
\end{array}} \right]$
        $ = \left[ {\begin{array}{*{20}{c}}
  {(1 \times 5) + (2 \times 7)}&{(1 \times 6) + (2 \times 8)} \\
  {(3 \times 5) + (4 \times 7)}&{(3 \times 6) + (4 \times 8)}
\end{array}} \right]$
        $ = \left[ {\begin{array}{*{20}{c}}
  {5 + 14}&{6 + 16} \\
  {15 + 28}&{18 + 32}
\end{array}} \right]$
$AB = \left[ {\begin{array}{*{20}{c}}
  {19}&{22} \\
  {43}&{50}
\end{array}} \right] \ldots \ldots \ldots \ldots \ldots eq(2)$

Now, $BA = \left[ {\begin{array}{*{20}{c}}
  5&6 \\
  7&8
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
  1&2 \\
  3&4
\end{array}} \right]$
$BA = \left[ {\begin{array}{*{20}{c}}
  {(5 \times 1) + (6 \times 3)}&{(5 \times 2) + (6 \times 4)} \\
  {(7 \times 1) + (8 \times 3)}&{(7 \times 2) + (8 \times 4)}
\end{array}} \right]$
        $ = \left[ {\begin{array}{*{20}{c}}
  {5 + 18}&{10 + 24} \\
  {7 + 24}&{14 + 32}
\end{array}} \right]$
$BA = \left[ {\begin{array}{*{20}{c}}
  {23}&{34} \\
  {31}&{46}
\end{array}} \right] \ldots \ldots \ldots \ldots \ldots eq(3)$

Here, $eq(2) \ne eq(3)$, that is, $AB \ne BA$
So, let’s check all the options:
Option A is not possible, as ${(A + B)^2} \ne {A^2} - 2AB + {B^2}$ $[\because $from $eq(1)]$
Option B is possible, as ${(A + B)^2} = {A^2} + 2AB + {B^2}$ $[\because $from $eq(1)]$
Option C is not possible, as $AB \ne BA$, so ${(A + B)^2} \ne {A^2} + 2BA + {B^2}$

Therefore, the correct option is (B).

Additional Information: In this type of question, where we have to expand a given expression and match it with the given options, we can do this using an alternate method by assuming two matrices, the matrix $A$ and matrix $B$, and then solve the given expression followed by solving all the expressions given in the options to check the final value of the expression mentioned in the question matches with which of the expressions mentioned in the options.

Note: Since, in this question, we are using algebraic identities and matrix multiplication together, so always keep in mind that unless and until it’s not mentioned in the question that $AB = BA$, it is always $AB \ne BA$, by default.