Understanding Singular Matrices and Their Adjoints

This solution explains the relationship between a singular matrix $A$ and its adjoint matrix, $\text{adj}(A)$, based on fundamental properties of matrices.

Key Concepts: Singular Matrix and Adjoint Matrix

Let's define the terms relevant to the question:

• Singular Matrix: A square matrix $A$ is called singular if its determinant is zero. Mathematically, this is represented as $ |A| = 0 $. The question specifies that $A$ is a singular matrix of order $ n \ge 2 $.
• Adjoint Matrix ($\text{adj}(A)$): The adjoint of a square matrix $A$ is the transpose of its cofactor matrix.
• Identity Matrix ($I$): A square matrix with ones on the main diagonal and zeros elsewhere.
• Null Matrix ($O$): A matrix where all elements are zero.

Fundamental Matrix Property

A crucial property connecting any square matrix $A$ and its adjoint $\text{adj}(A)$ is given by the formula:

$$ A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| \cdot I $$

where $ |A| $ is the determinant of matrix $A$, and $ I $ is the identity matrix of the same order.

Analyzing the Properties of a Singular Matrix

Since the given matrix $A$ is singular, we know that its determinant is zero:

$$ |A| = 0 $$

Evaluating the Options

Let's examine each statement using the property $ A \cdot \text{adj}(A) = |A| \cdot I $ and the condition $ |A| = 0 $.

Option 1: $A[\text{adj}(A)] = I$

Using the property $ A \cdot \text{adj}(A) = |A| \cdot I $, and substituting $ |A| = 0 $, we get:

$$ A \cdot \text{adj}(A) = 0 \cdot I = O $$

Since $ 0 \cdot I $ results in the null matrix ($O$), not the identity matrix ($I$), this statement is false.

Option 2: $|\text{adj}(A)| \ne 0$

There is a known property for the determinant of the adjoint matrix:

$$ |\text{adj}(A)| = |A|^{n-1} $$

Given that $ A $ is singular, $ |A| = 0 $. For the order $ n \ge 2 $, we have $ n-1 \ge 1 $. Therefore:

$$ |\text{adj}(A)| = 0^{n-1} = 0 $$

This means the determinant of the adjoint matrix is zero. The statement claims it is non-zero, so this statement is false.

Option 3: $\text{adj}(A)$ is a non-singular matrix

A matrix is non-singular if its determinant is non-zero. From the analysis of Option 2, we found that $ |\text{adj}(A)| = 0 $. A matrix with a determinant of zero is a singular matrix.

Therefore, $\text{adj}(A)$ is a singular matrix, not a non-singular one. This statement is false.

Option 4: $A[\text{adj}(A)] = O$ (Null matrix)

As derived in the analysis of Option 1, we use the fundamental property:

$$ A \cdot \text{adj}(A) = |A| \cdot I $$

Since $A$ is a singular matrix, $ |A| = 0 $. Substituting this value:

$$ A \cdot \text{adj}(A) = 0 \cdot I $$

Multiplying the scalar $0$ by the identity matrix $I$ results in the null matrix $O$.

$$ A \cdot \text{adj}(A) = O $$

This statement accurately describes the relationship for a singular matrix $A$ and its adjoint.

Conclusion

Based on the properties of determinants and the relationship between a matrix and its adjoint, the only true statement for a singular matrix $A$ of order $n \ge 2$ is that the product of the matrix and its adjoint equals the null matrix.