Given that A is a non-singular matrix of order 3 such that `A^2 = 2A`, then value of |2A| is:

Options

A. 4
B. 8
C. 64
D. 16

Answer

To find the value of |2A| given that A is a non-singular matrix of order 3 and satisfies the equation \( A^2 = 2A \), we can follow these steps:

### Step 1: Analyze the given equation
We start with the equation:
\[ A^2 = 2A \]
This can be rearranged to:
\[ A^2 - 2A = 0 \]

### Step 2: Factor the equation
Factoring the left-hand side gives us:
\[ A(A - 2I) = 0 \]
where \( I \) is the identity matrix of the same order as \( A \).

### Step 3: Use the property of non-singular matrices
Since \( A \) is a non-singular matrix, it cannot have any zero eigenvalues. Therefore, the equation \( A(A - 2I) = 0 \) implies that:
\[ A - 2I = 0 \quad \Rightarrow \quad A = 2I \]

### Step 4: Find 2A
Now, we can find \( 2A \):
\[ 2A = 2(2I) = 4I \]

### Step 5: Calculate the determinant of 2A
The determinant of a scalar multiple of the identity matrix can be calculated as follows:
\[ |2A| = |4I| = 4^3 = 64 \]
since the determinant of \( kI \) for an \( n \times n \) identity matrix is \( k^n \).

### Final Answer
Thus, the value of \( |2A| \) is:
\[ \boxed{64} \]