Analyzing Non-singular Matrix Adjoint Properties

Let A be a non-singular matrix of order 3. A matrix is non-singular if its determinant, denoted by \(|A|\), is non-zero (\(|A| \neq 0\)). The adjoint of a matrix A, denoted by adj A, is the transpose of the cofactor matrix of A.

We need to evaluate the correctness of the following two statements:

1. A (adj A) = (adj A) A
2. |adj A| = |A|

Evaluation of Statement 1: A (adj A) = (adj A) A

There is a fundamental property relating a square matrix, its adjoint, and its determinant. For any square matrix A of order n, the following relationship holds:

\(\text{A (adj A) = (adj A) A = |A| I}_n\)

where \(I_n\) is the identity matrix of order n. This property states that the product of a matrix and its adjoint (in either order) is equal to the determinant of the matrix times the identity matrix of the same order.

In this case, the matrix A is of order 3 (n=3), and it is non-singular (\(|A| \neq 0\)). The property still applies. Therefore, for the given matrix A:

\(\text{A (adj A) = (adj A) A = |A| I}_3\)

Since both A (adj A) and (adj A) A are equal to the same matrix \(|A| I_3\), it follows that A (adj A) = (adj A) A.

Thus, Statement 1 is correct for any square matrix, including a non-singular matrix of order 3.

Evaluation of Statement 2: |adj A| = |A|

There is a specific property relating the determinant of the adjoint of a square matrix to the determinant of the matrix itself. For a square matrix A of order n, the determinant of its adjoint is given by:

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

In this question, the matrix A is of order 3, so n=3. Applying the formula for n=3, we get:

\(|\text{adj A}| = |\text{A}|^{3-1} = |\text{A}|^2\)

So, for a matrix of order 3, the correct relationship is \(|\text{adj A}| = |\text{A}|^2\).

Statement 2 claims that \(|\text{adj A}| = |\text{A}|\). Comparing this with the correct relationship, we see that this statement is only true if \(|\text{A}|^2 = |\text{A}|\). This equation can be rewritten as \(|\text{A}|^2 - |\text{A}| = 0\), or \(|\text{A}|(|\text{A}| - 1) = 0\).

This equation is satisfied if \(|\text{A}| = 0\) or \(|\text{A}| = 1\).

However, the problem specifies that A is a non-singular matrix, which means \(|\text{A}| \neq 0\). Therefore, the only way for \(|\text{adj A}| = |\text{A}|\) to hold for a non-singular matrix of order 3 is if \(|\text{A}| = 1\).

Since the statement \(|\text{adj A}| = |\text{A}|\) is not true for all non-singular matrices of order 3 (it's only true when \(|A|=1\)), Statement 2 is not generally correct.

Conclusion

Based on the analysis:

• Statement 1 (A (adj A) = (adj A) A) is correct for any square matrix.
• Statement 2 (|adj A| = |A|) is incorrect for a general non-singular matrix of order 3, as the correct relationship is \(|\text{adj A}| = |\text{A}|^2\).

Therefore, only Statement 1 is correct.

Revision Table: Matrix Adjoint Properties

Additional Information on Non-singular Matrices and Adjoint

• Non-singular Matrix: A square matrix A is non-singular if its determinant is non-zero (\(|A| \neq 0\)). This is equivalent to saying that the matrix A has an inverse (\(A^{-1}\) exists).
• Inverse Matrix: For a non-singular matrix A, the inverse \(A^{-1}\) can be calculated using the adjoint: \(A^{-1} = \frac{1}{|A|} \text{adj A}\).
• Singular Matrix: A square matrix is singular if its determinant is zero (\(|A| = 0\)). A singular matrix does not have an inverse.
• Adjoint of a Singular Matrix: If A is a singular matrix of order n > 1, then \(A (\text{adj A}) = (adj A) A = |A| I_n = 0 \cdot I_n = 0\), the zero matrix. Also, for a singular matrix of order n > 1, \(|\text{adj A}| = |\text{A}|^{n-1} = 0^{n-1} = 0\). So, if A is singular and n > 1, its adjoint is also singular.