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Question: For \(3 \times 3\)matrices \(M\) and \(N\), which of the following statement(s) is (are) NOT correct...

For 3×33 \times 3matrices MM and NN, which of the following statement(s) is (are) NOT correct?
A. NTMN{N^T}MN is symmetric or skew-symmetric, according to as MM is symmetric or skew-symmetric.
B. MN−NMMN - NM is skew-symmetric for all symmetric matrices MM and NN
C. MNMN is symmetric for all symmetric matrices MM and NN
D. (adjM)(adjN)=adj(MN)\left( {adjM} \right)\left( {adjN} \right) = adj\left( {MN} \right) for all invertible matrices MM and NN

Explanation

Solution

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:
⇒BT=B\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:
⇒BT=−B\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 3×33 \times 3 matrices.
Now we have to check every option whether it is true.
A. To check whether NTMN{N^T}MN is symmetric or skew-symmetric, given M is symmetric or skew-symmetric.
Now first we find the transpose of NTMN{N^T}MN, as given below:
⇒(NTMN)T=NTMT(NT)T\Rightarrow {\left( {{N^T}MN} \right)^T} = {N^T}{M^T}{\left( {{N^T}} \right)^T}
⇒(NTMN)T=NTMN\Rightarrow {\left( {{N^T}MN} \right)^T} = {N^T}MN
As given M is symmetric matrix, hence MT=M{M^T} = M
∴(NTMN)T=NTMN\therefore {\left( {{N^T}MN} \right)^T} = {N^T}MN
Thus NTMN{N^T}MN is symmetric, so the first option is correct.
B. To check whether MN−NMMN - NM is skew-symmetric given thatMM and NN are symmetric matrices.
Now first we find the transpose of MN−NMMN - NM, as given below:
⇒(MN−NM)T=(MN)T−(NM)T\Rightarrow {\left( {MN - NM} \right)^T} = {\left( {MN} \right)^T} - {\left( {NM} \right)^T}
⇒(MN−NM)T=NTMT−MTNT\Rightarrow {\left( {MN - NM} \right)^T} = {N^T}{M^T} - {M^T}{N^T}
On simplifying the above expression, as given below:
⇒(MN−NM)T=NM−MN\Rightarrow {\left( {MN - NM} \right)^T} = NM - MN
⇒(MN−NM)T=−(MN−NM)\Rightarrow {\left( {MN - NM} \right)^T} = - \left( {MN - NM} \right)
Which means that MN−NMMN - NM is a skew-symmetric matrix.
Hence the second option is also correct.
C. MNMN is symmetric given the matrices MMand NNare symmetric.
⇒(MN)T=NTMT\Rightarrow {\left( {MN} \right)^T} = {N^T}{M^T}
⇒(MN)T=NM≠MN\Rightarrow {\left( {MN} \right)^T} = NM \ne MN
∴(MN)T≠MN\therefore {\left( {MN} \right)^T} \ne MN
Thus MNMN is not symmetric.
Hence the third option is not correct.
D. (adjM)(adjN)=adj(MN)\left( {adjM} \right)\left( {adjN} \right) = adj\left( {MN} \right) for all invertible matrices MM and NN
Here we know that the adjoint matrix of MN is given below:
⇒adj(MN)=(adjN)(adjM)\Rightarrow adj\left( {MN} \right) = \left( {adjN} \right)\left( {adjM} \right)
But (adjMN)≠(adjM)(adjN)\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 (AB)T{\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 (AB)T=BTAT{\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 (adjAB)=(adjB)(adjA)\left( {adjAB} \right) = \left( {adjB} \right)\left( {adjA} \right).