Adjugate Matrix Properties Analysis

This solution examines the properties of the adjugate matrix concerning the product of two non-singular matrices, \(A\) and \(B\), of the same order \(n\). We will evaluate three given statements based on fundamental principles of matrix algebra.

Matrix Properties Overview

• Non-Singular Matrix: A square matrix is non-singular if its determinant is non-zero. For such matrices, the inverse exists.
• Adjugate Matrix (\(\text{adj}(M)\)): The adjugate of a square matrix \(M\) is the transpose of its cofactor matrix.
• Fundamental Property: For any \(n \times n\) matrix \(M\), the product of the matrix and its adjugate is given by: \(M \cdot \text{adj}(M) = \text{adj}(M) \cdot M = |M| I_n\) where \(|M|\) is the determinant of \(M\) and \(I_n\) is the identity matrix of order \(n\).
• Determinant of a Product: For square matrices \(A\) and \(B\) of the same order, \(|AB| = |A||B|\).
• Adjugate of a Product: For non-singular matrices \(A\) and \(B\), the adjugate of their product is given by: \(\text{adj}(AB) = (\text{adj}B)(\text{adj}A)\) Note the reversal of the order compared to matrix multiplication.

Statement 1: Adjugate of Product Commutation

This statement proposes that the adjugate of the product \(AB\) is equal to the product of the adjugates \((\text{adj}A)(\text{adj}B)\).

However, the established property for the adjugate of a product of two matrices \(A\) and \(B\) is:

The statement \(\text{adj}(AB) = (\text{adj}A)(\text{adj}B)\) is only true if \((\text{adj}B)(\text{adj}A) = (\text{adj}A)(\text{adj}B)\), which means the adjugates of \(A\) and \(B\) must commute. This is not generally true for arbitrary non-singular matrices \(A\) and \(B\). Therefore, Statement 1 is incorrect.

Statement 2: Adjugate Product Equality

This statement claims that the adjugate of the product \(AB\) is equal to the adjugate of the product \(BA\). This implies:

As established previously, this equality requires that the adjugates of \(A\) and \(B\) commute. Since \(A\) and \(B\) are arbitrary non-singular matrices, their adjugates \((\text{adj}A)\) and \((\text{adj}B)\) do not necessarily commute. For example:

Let \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\).

We calculate:

• \(AB = \begin{pmatrix} 3 & 2 \\ 7 & 4 \end{pmatrix}\)
• \(BA = \begin{pmatrix} 1 & 2 \\ 4 & 6 \end{pmatrix}\)
• \(\text{adj}(A) = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}\)
• \(\text{adj}(B) = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}\)
• \(\text{adj}(AB) = \begin{pmatrix} 4 & -2 \\ -7 & 3 \end{pmatrix}\)
• \(\text{adj}(BA) = \begin{pmatrix} 6 & -2 \\ -4 & 1 \end{pmatrix}\)

Comparing \(\text{adj}(AB)\) and \(\text{adj}(BA)\), we see that \(\begin{pmatrix} 4 & -2 \\ -7 & 3 \end{pmatrix} \neq \begin{pmatrix} 6 & -2 \\ -4 & 1 \end{pmatrix}\).

Therefore, Statement 2 is incorrect.

Statement 3: Adjugate Property Verification

This statement relates to the fundamental property of the adjugate matrix.

For any square matrix \(M\), we know that \(M \cdot \text{adj}(M) = |M| I_n\). Let \(M = AB\). Since \(A\) and \(B\) are non-singular, their product \(AB\) is also non-singular, meaning \(|AB| = |A||B| \neq 0\).

Applying the property to the matrix \(AB\), we get:

Rearranging this equation, we obtain:

Here, \(0\) represents the null matrix (or zero matrix) of order \(n\). Thus, the expression \((AB)\text{adj}(AB) - |AB|I_n\) is indeed a null matrix.

Therefore, Statement 3 is correct.

Statements Correctness Summary

Evaluating the three provided statements regarding non-singular matrices \(A\) and \(B\):

• Statement 1: \(\text{adj}(AB) = (\text{adj}A)(\text{adj}B)\) - Incorrect.
• Statement 2: \(\text{adj}(AB) = \text{adj}(BA)\) - Incorrect.
• Statement 3: \((AB)\text{adj}(AB) - |AB|I_n\) is a null matrix - Correct.

Consequently, only one statement holds true.