`|adjA|=|A|^(2)` where A is a square matrix of order two.

Explanation

To prove the statement \(|\text{adj} A| = |A|^2\) for a square matrix \(A\) of order 2, we will follow these steps:

### Step 1: Understand the properties of the adjoint matrix
The adjoint (or adjugate) of a matrix \(A\), denoted as \(\text{adj} A\), is defined as the transpose of the cofactor matrix of \(A\). For any square matrix \(A\) of order \(n\), the determinant of the adjoint matrix can be expressed as:
\[ |\text{adj} A| = |A|^{n-1} \]
where \(n\) is the order of the matrix \(A\).

### Step 2: Apply the formula to a 2x2 matrix
Since \(A\) is a square matrix of order 2, we have \(n = 2\). Therefore, we can substitute \(n\) into the formula:
\[ |\text{adj} A| = |A|^{2-1} = |A|^1 = |A| \]

### Step 3: Compare with the given statement
The statement we need to verify is:
\[ |\text{adj} A| = |A|^2 \]
From our calculation, we found that:
\[ |\text{adj} A| = |A| \]
This implies:
\[ |A| = |A|^2 \]

### Step 4: Analyze the implications
The equation \(|A| = |A|^2\) can be rearranged to:
\[ |A|^2 - |A| = 0 \]
Factoring gives:
\[ |A|(|A| - 1) = 0 \]
This means that either \(|A| = 0\) or \(|A| = 1\).

### Conclusion
Thus, the statement \(|\text{adj} A| = |A|^2\) holds true only under specific conditions (i.e., when \(|A| = 0\) or \(|A| = 1\)). Therefore, the general statement is not valid for all 2x2 matrices.

### Final Result
The statement \(|\text{adj} A| = |A|^2\) is **false** for a general square matrix of order 2.

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