Question

For $3 \times 3$matrices $M$ and $N$, which of the following statement(s) is (are) NOT correct?
A. ${N^T}MN$ is symmetric or skew-symmetric, according to as $M$ is symmetric or skew-symmetric.
B. $MN - NM$ is skew-symmetric for all symmetric matrices $M$ and $N$
C. $MN$ is symmetric for all symmetric matrices $M$ and $N$
D. \[\left( {adjM} \right)\left( {adjN} \right) = adj\left( {MN} \right)\] for all invertible matrices $M$ and $N$

Answer 1

Hint: To solve this problem we need to have a basic knowledge about matrices. Just a few concepts regarding matrices are required such as transpose of a matrix, adjoint of a matrix, symmetric matrices and skew-symmetric matrices. If the transpose of a matrix is the same as the original matrix, it is a symmetric matrix, but whereas if it is the same as the negative of the original matrix, then it is a skew-symmetric matrix.

Complete step-by-step solution:
A matrix is a symmetric if, let suppose a matrix of named B. So matrix B is symmetric when:
$ \Rightarrow {B^T} = B$
That is, if the transpose of the matrix B is equal to the matrix B, then the matrix B is symmetric.
The matrix is said to be skew-symmetric, if:
$ \Rightarrow {B^T} = - B$
If the transpose of the matrix B is equal to the negative of matrix B, then the matrix B is skew-symmetric.
Given that there are two matrices named M and N. Both the matrices M and N are $3 \times 3$ matrices.
Now we have to check every option whether it is true.
A. To check whether ${N^T}MN$ is symmetric or skew-symmetric, given M is symmetric or skew-symmetric.
Now first we find the transpose of ${N^T}MN$, as given below:
\[ \Rightarrow {\left( {{N^T}MN} \right)^T} = {N^T}{M^T}{\left( {{N^T}} \right)^T}\]
\[ \Rightarrow {\left( {{N^T}MN} \right)^T} = {N^T}MN\]
As given M is symmetric matrix, hence ${M^T} = M$
$\therefore {\left( {{N^T}MN} \right)^T} = {N^T}MN$
Thus \[{N^T}MN\] is symmetric, so the first option is correct.
B. To check whether $MN - NM$ is skew-symmetric given that$M$ and $N$ are symmetric matrices.
 Now first we find the transpose of $MN - NM$, as given below:
$ \Rightarrow {\left( {MN - NM} \right)^T} = {\left( {MN} \right)^T} - {\left( {NM} \right)^T}$
$ \Rightarrow {\left( {MN - NM} \right)^T} = {N^T}{M^T} - {M^T}{N^T}$
On simplifying the above expression, as given below:
$ \Rightarrow {\left( {MN - NM} \right)^T} = NM - MN$
$ \Rightarrow {\left( {MN - NM} \right)^T} = - \left( {MN - NM} \right)$
Which means that $MN - NM$ is a skew-symmetric matrix.
Hence the second option is also correct.
C. $MN$ is symmetric given the matrices $M$and $N$are symmetric.
$ \Rightarrow {\left( {MN} \right)^T} = {N^T}{M^T}$
$ \Rightarrow {\left( {MN} \right)^T} = NM \ne MN$
$\therefore {\left( {MN} \right)^T} \ne MN$
Thus $MN$ is not symmetric.
Hence the third option is not correct.
D. \[\left( {adjM} \right)\left( {adjN} \right) = adj\left( {MN} \right)\] for all invertible matrices $M$ and $N$
Here we know that the adjoint matrix of MN is given below:
$ \Rightarrow adj\left( {MN} \right) = \left( {adjN} \right)\left( {adjM} \right)$
But $\left( {adjMN} \right) \ne \left( {adjM} \right)\left( {adjN} \right)$

Option A and B are the correct options.

Note: Here while solving the transpose of a product of two matrices named A and B, which is given by ${\left( {AB} \right)^T}$, it is given by the product of the transpose of the matrix B first and then the transpose of the matrix A, which is given by ${\left( {AB} \right)^T} = {B^T}{A^T}$. This case is similar while finding the adjoint of a product of two matrices which is given by $\left( {adjAB} \right) = \left( {adjB} \right)\left( {adjA} \right)$.