Let A be a non-singular square matrix of order n. Then; `|adjA| =

Explanation

To find the value of \(|\text{adj} A|\) where \(A\) is a non-singular square matrix of order \(n\), we can follow these steps:

### Step 1: Understand the relationship between a matrix and its adjoint
The adjoint (or adjugate) of a matrix \(A\), denoted as \(\text{adj} A\), is defined such that:
\[ A \cdot \text{adj} A = |\text{det} A| \cdot I_n \]
where \(I_n\) is the identity matrix of order \(n\).

### Step 2: Take the determinant of both sides
Taking the determinant of both sides of the equation gives us:
\[ |\text{det}(A \cdot \text{adj} A)| = |\text{det}(|\text{det} A| \cdot I_n)| \]

### Step 3: Apply the property of determinants
Using the property of determinants, we have:
\[ |\text{det}(A) \cdot \text{det}(\text{adj} A)| = |\text{det} A|^n \]
This is because the determinant of a scalar multiple of the identity matrix is the scalar raised to the power of the order of the matrix.

### Step 4: Substitute and simplify
Let \(d = |\text{det} A|\). Then we can rewrite the equation as:
\[ d \cdot |\text{det}(\text{adj} A)| = d^n \]
Now, dividing both sides by \(d\) (since \(A\) is non-singular, \(d \neq 0\)):
\[ |\text{det}(\text{adj} A)| = d^{n-1} \]

### Step 5: Conclusion
Thus, we conclude that:
\[ |\text{adj} A| = |\text{det} A|^{n-1} \]

### Final Answer
\[ |\text{adj} A| = |\text{det} A|^{n-1} \]

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