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Question: Let A and B be any two \( 3 \times 3 \) matrices . If A is symmetric and B is skew symmetric, then t...

Let A and B be any two 3×33 \times 3 matrices . If A is symmetric and B is skew symmetric, then the matrix AB - BA is

Explanation

Solution

In order to solve the given question, we must have the knowledge of matrices and the concept of the terms used here Symmetric and Skew symmetric . We can differentiate between the terms by their properties . The transpose of a symmetric matrix is always the same given as it is but the transpose of Skew symmetric is the negative of the matrix given . Also we will be using the Reversal Rule which states that P and Q are two matrix of same order then product of (PQ)T{\left( {PQ} \right)^T} will be defined as QTPT{Q^T}{P^T} where superscript T is the symbol of transpose . Now as we know about these rules we can easily solve this question .

Complete step-by-step solution:
In the question above we are given that matrix A is symmetric and as per the concept The transpose of the symmetric matrix is always the same given as it is . So, we get –
⇒AT=A\Rightarrow {A^T} = A
Also we are given that the B is a Skew symmetric matrix which has the property as the transpose of Skew symmetric is the negative of the matrix given . So, we get –
⇒BT=−B\Rightarrow {B^T} = - B
Also we will be using the Reversal Rule which states that P and Q are two matrix of same order then product of (PQ)T{\left( {PQ} \right)^T} will be defined as QTPT{Q^T}{P^T} where superscript T is the symbol of transpose
⇒(PQ)T\Rightarrow {\left( {PQ} \right)^T} = QTPT{Q^T}{P^T}
Now, we need to find the nature of AB – BA matrix so let us take the transpose of the matrix AB – BA
⇒(AB − BA)T=(AB)T−(BA)T\Rightarrow {(AB{\text{ }}-{\text{ }}BA)^T} = {(AB)^T} - {(BA)^T}
Now since it involves the product of matrices of the same order, we will now use here Reversal Rule as stated above .

⇒(AB − BA)T=(AB)T−(BA)T ⇒(AB − BA)T=BTAT−ATBT  \Rightarrow {(AB{\text{ }}-{\text{ }}BA)^T} = {(AB)^T} - {(BA)^T} \\\ \Rightarrow {(AB{\text{ }}-{\text{ }}BA)^T} = {B^T}{A^T} - {A^T}{B^T} \\\

Now as it is given that AT=A{A^T} = A and BT=−B{B^T} = - B, so we will substitute the values in place of this, we get –

⇒(AB − BA)T=(AB)T−(BA)T ⇒(AB − BA)T=BTAT−ATBT ⇒(AB − BA)T=−BA−A(−B) ⇒(AB − BA)T=−BA+AB ⇒(AB − BA)T=AB−BA \Rightarrow {(AB{\text{ }}-{\text{ }}BA)^T} = {(AB)^T} - {(BA)^T} \\\ \Rightarrow {(AB{\text{ }}-{\text{ }}BA)^T} = {B^T}{A^T} - {A^T}{B^T} \\\ \Rightarrow {(AB{\text{ }}-{\text{ }}BA)^T} = - BA - A( - B) \\\ \Rightarrow {(AB{\text{ }}-{\text{ }}BA)^T} = - BA + AB \\\ \Rightarrow {(AB{\text{ }}-{\text{ }}BA)^T} = AB - BA \\\

And now we know AB - BA is symmetric and as per the concept The transpose of a symmetric matrix is always the same given as it is .
(AB − BA)T=AB−BA{(AB{\text{ }}-{\text{ }}BA)^T} = AB - BA
This shows that AB – BA is symmetric in nature .

Therefore, the correct answer is option ‘B’ .

Note: Always try to understand the mathematical statement carefully and keep things distinct .
Remember the properties like here we applied one property to make the other property proved so apply appropriately. Try to solve the question using the diagram which makes it visual and easy to understand. Cross check the answer and always keep the final answer simplified .