Animated Solution for Mathematics - Matrices and Determinants: For $3\times 3$ matrices $M$ and $N$, which of the following statement(s) is (are) NOT correct?

The Reversal Law

Before we touch the options, we must anchor ourselves in the Reversal Law. In scalar arithmetic, $ab=ba$. But in the matrix world, order is everything.

The Reversal Law states that for any two matrices $A$ and $B$, the transpose of their product is the product of their transposes in reverse order:

This is not just a rule; it is a geometric necessity. When you transpose a product, you are essentially flipping the entire operation, which forces the matrices to swap their positions. This law is the key to unlocking every single option in this problem.

The Symmetry of $N^{T}MN$

Let us define a matrix $X=N^{T}MN$. To determine if it is symmetric or skew-symmetric, we take its transpose:

Applying the Reversal Law, we get:

Since the transpose of a transpose is the original matrix, $(N^T{)}^T=N$. Thus, we arrive at:

Now, consider the two cases. If $M$ is symmetric, $M^T=M$, so $X^T=N^{T}MN=X$. If $M$ is skew-symmetric, $M^T=-M$, so $X^T=N^T(-M)N=-X$. The nature of $X$ is perfectly mirrored by the nature of $M$. Option A is correct.

The Skew-Symmetry of $MN-NM$

Now, consider $Y=MN-NM$, where $M$ and $N$ are symmetric. Taking the transpose:

Using the Reversal Law, we get:

Since $M$ and $N$ are symmetric, $M^T=M$ and $N^T=N$, so:

Factoring out the negative sign, we get $Y^T=-(MN-NM)=-Y$. This is the definition of a skew-symmetric matrix. Option B is correct.

Option C

Here is where the JEE examiners love to set a trap. Option C claims that $MN$ is symmetric if $M$ and $N$ are symmetric. Let us test this:

For $MN$ to be symmetric, we need $(MN{)}^T=MN$, which implies $NM=MN$. But we know that matrix multiplication is not commutative in general. Unless $M$ and $N$ commute, $MN$ is not necessarily symmetric. This statement is incorrect.

Option D

Finally, let us look at the adjoint property. The adjoint of a product follows the same Reversal Law as the transpose and the inverse:

The statement in Option D claims $(\text{adj }M)(\text{adj }N)=\text{adj}(MN)$. This is only true if the adjoints commute, which is not guaranteed. Therefore, this statement is also incorrect.

The Takeaway

We have successfully navigated the traps. The key takeaways are:

1. Matrix multiplication is not commutative ($MN eq NM$). 2. The Reversal Law is universal for transposes, inverses, and adjoints:

By keeping these principles in mind, you can dismantle any matrix problem with confidence. Keep practicing, and keep falling in love with the logic!