If `A=[[3,-3,4],[2,-3,4],[0,-1,1]]` , then

Options

A. `adj(adjA)=A`
B. `abs(adj(adj(A)))=1`
C. `abs(adj(A))=1`
D. None of these

Explanation

To solve the problem, we need to find the determinant of the given matrix \( A \) and then use the properties of determinants and adjoints to derive the results.

Given matrix:
\[ A = \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix} \]

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 3x3 matrix can be calculated using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is represented as:
\[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]

For our matrix \( A \):
- \( a = 3, b = -3, c = 4 \)
- \( d = 2, e = -3, f = 4 \)
- \( g = 0, h = -1, i = 1 \)

Substituting these values into the determinant formula:
\[ \text{det}(A) = 3((-3)(1) - (4)(-1)) - (-3)((2)(1) - (4)(0)) + 4((2)(-1) - (-3)(0)) \]

Calculating each term:
1. \( (-3)(1) - (4)(-1) = -3 + 4 = 1 \)
2. \( (2)(1) - (4)(0) = 2 - 0 = 2 \)
3. \( (2)(-1) - (-3)(0) = -2 - 0 = -2 \)

Now substituting these back into the determinant equation:
\[ \text{det}(A) = 3(1) - (-3)(2) + 4(-2) \]
\[ = 3 + 6 - 8 \]
\[ = 1 \]

### Step 2: Use the Property of Adjoint
According to the property of adjoints:
\[ \text{adj}(\text{adj}(A)) = \text{det}(A)^{n-2} A \]
where \( n \) is the order of the matrix. Here, \( n = 3 \), so:
\[ \text{adj}(\text{adj}(A)) = \text{det}(A)^{3-2} A = \text{det}(A) A = 1 \cdot A = A \]

### Step 3: Verify Other Properties
1. **Property 1**: \(\text{adj}(A) = \text{det}(A)^{n-1} I\)
- For \( n = 3 \): \(\text{adj}(A) = \text{det}(A)^{2} I = 1^2 I = I\)

2. **Property 2**: \(\text{det}(\text{adj}(A)) = \text{det}(A)^{n-1}\)
- For \( n = 3 \): \(\text{det}(\text{adj}(A)) = \text{det}(A)^{2} = 1^2 = 1\)

### Conclusion
All the properties hold true, and we have verified that:
- \(\text{det}(A) = 1\)
- \(\text{adj}(\text{adj}(A)) = A\)
- \(\text{adj}(A) = I\)

Thus, all three options provided in the question are correct.