If A= `[{:(1,0,3),(2,1,1),(0,0,2):}]` , then the value of |adj(adj A) | is

Options

A. 14
B. 16
C. 15
D. 12

Answer

To find the value of |adj(adj A)| for the given matrix A, we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
Given matrix A is:
\[ A = \begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & 1 \\ 0 & 0 & 2 \end{pmatrix} \]

We will calculate the determinant of A using the formula for the determinant of a 3x3 matrix:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Where:
- \( a = 1, b = 0, c = 3 \)
- \( d = 2, e = 1, f = 1 \)
- \( g = 0, h = 0, i = 2 \)

Calculating:
\[ |A| = 1(1 \cdot 2 - 1 \cdot 0) - 0(2 \cdot 2 - 1 \cdot 0) + 3(2 \cdot 0 - 1 \cdot 0) \]
\[ = 1(2) - 0 + 3(0) \]
\[ = 2 \]

### Step 2: Use the Property of the Determinant of the Adjoint
The property of the determinant of the adjoint of a matrix states:
\[ |adj(A)| = |A|^{n-1} \]
where \( n \) is the order of the matrix. Here, \( n = 3 \).

Thus,
\[ |adj(A)| = |A|^{3-1} = |A|^2 = 2^2 = 4 \]

### Step 3: Calculate the Determinant of the Adjoint of the Adjoint
Now we need to find |adj(adj A)|. Using the same property:
\[ |adj(adj A)| = |adj(A)|^{n-1} \]
Here, \( n = 3 \) again.

Thus,
\[ |adj(adj A)| = |adj(A)|^{3-1} = |adj(A)|^2 = 4^2 = 16 \]

### Final Answer
The value of |adj(adj A)| is \( 16 \).

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