Understanding Non-Singular Matrices and Adjoints

This question asks about the properties of the product of a non-singular matrix \(A\) and its adjoint, \(B = \text{adj } A\). To solve this, we need to recall the fundamental relationship between a matrix, its adjoint, and its determinant.

Key Relationship: Matrix, Adjoint, and Determinant

For any square matrix \(A\) of order \(n\), the product of the matrix and its adjoint is given by the formula:

where \(\det A\) is the determinant of matrix \(A\), and \(I_n\) is the identity matrix of order \(n\). This relationship is crucial for analyzing the given statements.

Given that \(A\) is a non-singular matrix, its determinant \(\det A \neq 0\). Also, we are given \(B = \text{adj } A\). Substituting \(B\) into the relationship, we get:

Now let's examine each statement based on this result.

Analyzing Statement 1: AB = BA

Statement 1 says \(AB = BA\). From the fundamental relationship \(A(\text{adj } A) = (\text{adj } A)A\), and substituting \(B = \text{adj } A\), we directly get \(AB = BA\). This shows that the product of a matrix and its adjoint is commutative.

Therefore, Statement 1 is correct.

Analyzing Statement 2: AB is a scalar matrix

Statement 2 says \(AB\) is a scalar matrix. We found that \(AB = (\det A)I\). Let's understand what a scalar matrix is.

A scalar matrix is a diagonal matrix where all the diagonal entries are equal. The identity matrix \(I\) is a diagonal matrix with all diagonal entries equal to 1. When we multiply \(I\) by a scalar value (\(\det A\) in this case), we get a matrix where the diagonal entries are all equal to that scalar value (\(\det A\)), and off-diagonal entries are zero.

For example, if \(A\) is a 3x3 matrix, \(I\) is \(I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\). Then \((\det A)I_3 = \begin{pmatrix} \det A & 0 & 0 \\ 0 & \det A & 0 \\ 0 & 0 & \det A \end{pmatrix}\). This matrix fits the definition of a scalar matrix.

Since \(A\) is non-singular, \(\det A \neq 0\), so \((\det A)I\) is indeed a scalar matrix with diagonal elements equal to \(\det A\).

Therefore, Statement 2 is correct.

Analyzing Statement 3: AB can be a null matrix

Statement 3 says \(AB\) can be a null matrix. We know that \(AB = (\det A)I\). For \(AB\) to be a null matrix (a matrix with all entries being zero), we would need \((\det A)I = 0\).

Since \(I\) is the identity matrix, its diagonal entries are 1 (non-zero). The only way for \((\det A)I\) to be the zero matrix is if the scalar multiplier \(\det A\) is zero.

However, the question explicitly states that \(A\) is a non-singular matrix. By definition, a non-singular matrix has a non-zero determinant, i.e., \(\det A \neq 0\).

Since \(\det A \neq 0\), \((\det A)I\) cannot be the zero matrix (null matrix).

Therefore, Statement 3 is incorrect.

Summary of Statements

Based on our analysis, Statement 1 and Statement 2 are correct, while Statement 3 is incorrect.

The option that includes only statements 1 and 2 is the correct answer.

Revision Table: Non-singular Matrix and Adjoint Properties

Additional Information: Matrix Inverse

The relationship \(A(\text{adj } A) = (\text{adj } A)A = (\det A)I\) is also used to find the inverse of a non-singular matrix. Since \(\det A \neq 0\), we can divide by \(\det A\):

Similarly, \(\left(\frac{1}{\det A} \text{adj } A\right) A = I\). By the definition of matrix inverse, if \(AC = CA = I\), then \(C = A^{-1}\).

Thus, for a non-singular matrix \(A\), its inverse \(A^{-1}\) is given by:

This further highlights the importance of the determinant being non-zero for the inverse (and thus the adjoint relationship \(A(\text{adj } A) = (\det A)I\) to imply a non-null result) to exist in this context.