Question

If $A = \left[ {\begin{array}{*{20}{c}}
  0&1&{ - 2} \\
  { - 1}&0&5 \\
  2&{ - 5}&0
\end{array}} \right]$ , then which of the following options is correct?
A. $A' = A$
B. $A' = - A$
C. $A' = 2A$
D. None of these

Answer 1

Hint: Transpose of a matrix is calculated by interchanging either the rows with the columns or the columns with the rows. It is denoted by $X'$. Calculate the transpose of the matrix $A$ given in the above question and find out which option is correct regarding $A'$.

Complete step by step Solution:
Given matrix:
$A = \left[ {\begin{array}{*{20}{c}}
  0&1&{ - 2} \\
  { - 1}&0&5 \\
  2&{ - 5}&0
\end{array}} \right]$
We know that the Transpose of a matrix is calculated by interchanging either the rows with the columns or the columns with the rows.
Therefore, calculating its transpose,
$A' = \left[ {\begin{array}{*{20}{c}}
  0&{ - 1}&2 \\
  1&0&{ - 5} \\
  { - 2}&5&0
\end{array}} \right]$ … (1)
Clearly, $A' \ne A$ .
Now, let us calculate the value of $ - A$.
$ - A = \left[ {\begin{array}{*{20}{c}}
  0&{ - 1}&2 \\
  1&0&{ - 5} \\
  { - 2}&5&0
\end{array}} \right]$ … (2)
From (1) and (2),
$A' = - A$
Hence, for the given matrix $A$ , $A' = - A$ .

Hence, the correct option is (B).

Note: Two matrices $A$ and $B$ are said to be equal to each other when they both are of the same order and when every element of matrix $A$ is equal to the corresponding elements of matrix $B$.