Question

For \(3 \times 3\) matrices \(M\) and \(N\), which of the following statement(s) is (are) NOT correct? (A) \(N^{T} M N\) is symmetric or skew symmetric, according as \(M\) is symmetric or skew symmetric (B) \(M N-N M\) is skew symmetric for all symmetric matrices \(M\) and \(N\) (C) \(M N\) is symmetric for all symmetric matrices \(M\) and \(N\) (D) \((\operatorname{adj} M)(\operatorname{adj} N)=a d j(M N)\) for all invertible matrices \(M\) and \(N\)

Short answer

Statements C and D are NOT correct.

Step-by-step solution

Analyzing Statement A

To prove or disprove that the product of the form \(N^{T} M N\) has the same symmetry as M, one should remember that for any square matrix \(A\), \(A\) is symmetric if \(A = A^{T}\), and is skew-symmetric if \(A = -A^{T}\). If M is symmetric, then \(M^T = M\); hence \((N^T M N)^T = N^T M^T (N^T)^T = N^T M N\). Similarly, if M is skew-symmetric, then \(M^T = -M\); hence \((N^T M N)^T = N^T (-M) (N^T)^T = -N^T M N\). This confirms that statement A is correct.

Examining Statement B

We know that if M and N are symmetric, then \(M = M^T\) and \(N = N^T\). The product \(M N - N M\) can be evaluated and transposed to check for skew-symmetry; \((MN - NM)^T = N^T M^T - M^T N^T = NM - MN = -(MN - NM)\), which shows that \(MN - NM\) is skew-symmetric if M and N are symmetric. Therefore, statement B is also correct.

Checking Statement C

For \(M N\) to be symmetric given that M and N are symmetric, we should have \((MN)^T = N^T M^T = NM = MN\), but this only happens if M and N commute i.e., \(MN = NM\), which is not generally true for matrices. Therefore, the statement that \(M N\) is symmetric for all symmetric matrices \(M\) and \(N\) is not necessarily correct. Statement C is not correct.

Validating Statement D

For the adjugate property, general matrix properties tell us that \((adj(A))(adj(B)) eq adj(AB)\) in general. The correct relation involves the determinants of the matrices and is given by \((adj(A))(adj(B)) = (det(A))^{n-1}(adj(B))\) for n by n matrices, where n is 3 in this case. This implies that statement D is not correct, because it falsely claims an equality.

Key concepts

Symmetric Matrices

In the study of linear algebra, symmetric matrices have a special role due to their innate properties and applications. Imagine looking in a mirror; what you see is a reflection that mimics your movements. Symmetric matrices are somewhat like that in the world of matrices. Specifically, a symmetric matrix is defined as a square matrix that is equal to its transpose, so we have the mathematical relationship expressed as \( A = A^T \).

What this means is that if you flip a symmetric matrix over its main diagonal (the diagonal from the top-left to the bottom-right corner), you'll end up with the same matrix. If you're visual, picture swapping the elements across this diagonal, and every element falls into place perfectly. The elegance of symmetric matrices isn't just aesthetic; it's functional. For instance, they entail simpler computations when it comes to eigenvalues and can be diagonalized using an orthonormal basis.

Skew-Symmetric Matrices

Taking the concept of symmetry to a 'twisted' direction, skew-symmetric matrices, or antisymmetric matrices, are square matrices that are equal to the negative of their transpose. This means if you have a skew-symmetric matrix \( B \), it satisfies the condition \( B = -B^T \).

It's a bit like having a dance partner who always does the opposite of what you do, symmetric yet opposite. Skew-symmetric matrices have all zeroes on their main diagonal, since those elements are their own negatives. This property comes with its perks; for instance, it simplifies solving systems of linear equations when they appear in certain contexts. When dealing with physical systems, like the forces acting on an object, skew-symmetric matrices often pop up, emphasizing their real-world relevance.

Matrix Transpose Properties

The transpose of a matrix is like taking a matrix, --> giving it a spin <-- and flipping it along its diagonal. This isn't just a neat magic trick; it has significant implications in matrix algebra. Let's sprinkle some magic dust and look at some key properties. For any matrices \( A \) and \( B \) of appropriate dimensions:

• The transpose of a transpose brings you back home: \( (A^T)^T = A \).
• Transposing a sum? Just transpose and add: \( (A+B)^T = A^T+B^T \).
• What about multiplication? Swap the order: \( (AB)^T = B^T A^T \).

Understanding the properties of matrix transpose is like having a Swiss army knife for linear algebra—it multiplies your problem-solving tools.

Adjugate of a Matrix

The adjugate, or classical adjoint, of a matrix provides a whimsical twist by playing a role in finding the inverse of a matrix. Given a square matrix \( C \), its adjugate is formed by taking the transpose of its cofactor matrix. Now, the cofactor matrix is like a treasure map, filled with little 'cofactor' treasures calculated from the matrix.

But why should we care about the adjugate? Well, it dances beautifully with another matrix concept: the determinant. Specifically, for an invertible matrix \( C \), we have the relationship \( C^{-1} = \frac{1}{det(C)} adj(C) \). Sadly, the statement \((adj(M))(adj(N)) = adj(MN)\) uncovered in the exercise is not like our other magical properties; it doesn't always hold, which might be a bit disappointing for an eager mathematician.