If a is a square matrix of order 3 such that ` A^2 = 2A ` then find the value of |A|

Answer

To solve the problem, we start with the given equation for the square matrix \( A \):

\[ A^2 = 2A \]

We need to find the determinant of the matrix \( A \), denoted as \( |A| \).

### Step 1: Rewrite the equation
We can rearrange the equation \( A^2 - 2A = 0 \) to factor it:

\[ A^2 - 2A = A(A - 2I) = 0 \]

where \( I \) is the identity matrix of the same order as \( A \).

### Step 2: Apply the determinant
Taking the determinant on both sides of the equation gives us:

\[ |A(A - 2I)| = |0| \]

Since the determinant of the zero matrix is zero, we have:

\[ |A| \cdot |A - 2I| = 0 \]

### Step 3: Analyze the product
The product \( |A| \cdot |A - 2I| = 0 \) implies that at least one of the factors must be zero. Thus, we have two cases:

1. \( |A| = 0 \)
2. \( |A - 2I| = 0 \)

### Step 4: Consider the first case
If \( |A| = 0 \), then \( A \) is singular, which means that the determinant of \( A \) is zero.

### Step 5: Consider the second case
If \( |A - 2I| = 0 \), this means that \( A - 2I \) is also singular, indicating that \( 2 \) is an eigenvalue of the matrix \( A \).

### Step 6: Conclusion
Since we have established that either \( |A| = 0 \) or \( |A - 2I| = 0 \), we conclude that the determinant of \( A \) must be zero.

Thus, the value of \( |A| \) is:

\[ \boxed{0} \]