Question

If A= \[\left[
{\begin{array}{*{20}{c}}1&{ - 2}\\5&3\end{array}} \right]\]then \[A + {A^T}\]equals
A. \[\left[
{\begin{array}{*{20}{c}}2&3\\3&6\end{array}} \right]\]
B. \[\left[
{\begin{array}{*{20}{c}}2&{ - 4}\\{10}&6\end{array}} \right]\]
C. \[\left[
{\begin{array}{*{20}{c}}2&4\\{ - 10}&6\end{array}} \right]\]
D. None of these

Answer 1

Hint:
We are given a 2×2 matrix \[\left[
{\begin{array}{*{20}{c}}1&{ - 2}\\5&3\end{array}} \right]\]and
We have to first find its transpose and then by adding these two matrices
which are, given matrix and transpose matrix, we get our required solution. As
2×2 matrix will add to its transpose will give us 2×2 matrix only.





Formula Used:
Let us assume we have a matrix A=\[\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\\{{b_1}}&{{b_2}}\end{array}} \right]\].
So for finding its transpose we have to interchange rows to
columns and columns to rows.
The we will get \[{A^T} = \left[{\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}\\{{a_2}}&{{b_2}}\end{array}}\right]\]




Complete step-by-step answer:
We have given a matrix A=\[\left[ {\begin{array}{*{20}{c}}1&{ - 2}\\5&3\end{array}} \right]\] and
firstly we have to find its transpose and then add it to the given matrix.
So of the given matrix \[\left[
{\begin{array}{*{20}{c}}1&{ - 2}\\5&3\end{array}} \right]\]is
\[{A^T} = \left[
{\begin{array}{*{20}{c}}1&5\\{ - 2}&3\end{array}} \right]\] as we
have interchange row to column and column to row. This is the formula for
transpose of a 2×2 matrix.
Now we have to find \[A+ {A^T}\]
\[A + {A^T} = \left[{\begin{array}{*{20}{c}}1&{ - 2}\\5&3\end{array}} \right] + \left[{\begin{array}{*{20}{c}}1&5\\{ - 2}&3\end{array}} \right]\]
\[A + {A^T} = \left[{\begin{array}{*{20}{c}}2&3\\3&6\end{array}} \right]\]
Hence by adding two 2×2 matrices which are the matrix which we are
given and its transpose, we will get our required solution as a 2×2 matrix.
Hence option A is correct.

Note:
We have some properties of transpose which we have to remember so
that it will be easy for students while solving such questions.
If we take the transpose of the transpose matrix,
the matrix obtained is equal to the original matrix. Hence, for a matrix A, (A’)’ = A
Transpose of an addition of two matrices A and
B obtained will be exactly equal to the sum of the transpose of individual
matrices A and B. This means,
\[{(A + B)^T} = {A^T}+ {B^T}\]
If a matrix is multiplied by a constant and its transpose is
taken, then the matrix obtained is equal to the transpose of the original
matrix multiplied by that constant. That is,
\[{(kA)^T}= k{A^T}\]