Question

Let A and B be any two \[3 \times 3\] matrices. If A is symmetric and B is skew symmetric, then the matrix \[AB - BA\] is:
A) Skew symmetric
B) Symmetric
C) Neither symmetric nor skew symmetric
D) I or –I, where I is the identity matrix

Answer 1

Hint: Here we will find the transpose of the given matrix and then use the concept of symmetric and skew symmetric matrix i.e.
If a matrix X is symmetric then \[{\left( X \right)^T} = X\]
If a matrix Y is skew symmetric then \[{\left( Y \right)^T} = - Y\]

Complete step-by-step answer:
The given matrix is:-
\[AB - BA\]
Taking transpose of the above matrix we get:-
\[{\left( {AB - BA} \right)^T} = {\left( {AB} \right)^T} - {\left( {BA} \right)^T}\]
Now we know that:
\[{\left( {XY} \right)^T} = {Y^T}{X^T}\]
Hence, applying this property we get:-
\[{\left( {AB - BA} \right)^T} = {B^T}{A^T} - {A^T}{B^T}\]……………………………………….(1)
Now since A is symmetric matrix
Therefore, \[{A^T} = A\]
Since B is skew symmetric matrix
Therefore,
\[{B^T} = - B\]
Hence substituting the values in equation 1 we get:-
\[{\left( {AB - BA} \right)^T} = \left( { - B} \right)\left( A \right) - \left( A \right)\left( { - B} \right)\]
Simplifying it further we get:-
\[\begin{gathered}
  {\left( {AB - BA} \right)^T} = - BA + AB \\
   \Rightarrow {\left( {AB - BA} \right)^T} = AB - BA \\
\end{gathered} \]
Hence, \[AB - BA\] is a symmetric matrix.

Therefore, option A is the correct option.

Note: Students should note that only the square matrices can be symmetric or skew-symmetric form.
Also, matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A and the upper triangular matrix is equal to the lower triangular matrix
\[\left[ {\begin{array}{*{20}{c}}
  a&b&c \\
  b&d&f \\
  c&f&e
\end{array}} \right]\]
Matrix A is said to be skew-symmetric if the transpose of matrix A is equal to negative of matrix A and the upper triangular matrix is negative to the lower triangular matrix or vice-versa.
\[\left[ {\begin{array}{*{20}{c}}
  a&b&c \\
  { - b}&d&f \\
  { - c}&{ - f}&e
\end{array}} \right]\]