Let `A=[(1,3cos 2theta,1),(sin2theta, 1, 3 cos 2 theta),(1, sin 2 theta, 1)]` the maximum value of `|A|` is equal to k, then `(k-3)^(2)` is equal to

Answer

To solve the problem, we need to find the determinant of the given matrix \( A \) and then determine its maximum value. Let's go through the steps systematically.

### Step 1: Write down the matrix
The matrix \( A \) is given as:
\[ A = \begin{pmatrix} 1 & 3\cos(2\theta) & 1 \\ \sin(2\theta) & 1 & 3\cos(2\theta) \\ 1 & \sin(2\theta) & 1 \end{pmatrix} \]

### Step 2: Calculate the determinant \( |A| \)
We will use the determinant formula for a \( 3 \times 3 \) matrix:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where:
- \( a = 1 \), \( b = 3\cos(2\theta) \), \( c = 1 \)
- \( d = \sin(2\theta) \), \( e = 1 \), \( f = 3\cos(2\theta) \)
- \( g = 1 \), \( h = \sin(2\theta) \), \( i = 1 \)

Calculating the determinant:
\[ |A| = 1 \cdot (1 \cdot 1 - 3\cos(2\theta) \cdot \sin(2\theta)) - 3\cos(2\theta) \cdot (\sin(2\theta) \cdot 1 - 3\cos(2\theta) \cdot 1) + 1 \cdot (\sin(2\theta) \cdot \sin(2\theta) - 1 \cdot 1) \]

This simplifies to:
\[ |A| = 1 - 3\cos(2\theta)\sin(2\theta) - 3\cos(2\theta)(\sin(2\theta) - 3\cos(2\theta)) + (\sin^2(2\theta) - 1) \]

### Step 3: Simplify the expression
Now we simplify:
\[ |A| = 1 - 3\cos(2\theta)\sin(2\theta) - 3\cos(2\theta)\sin(2\theta) + 9\cos^2(2\theta) + \sin^2(2\theta) - 1 \]
\[ |A| = 9\cos^2(2\theta) + \sin^2(2\theta) - 6\cos(2\theta)\sin(2\theta) \]

### Step 4: Use trigonometric identities
Using the identity \( \sin^2(2\theta) + \cos^2(2\theta) = 1 \):
\[ |A| = 8\cos^2(2\theta) + 1 - 6\cos(2\theta)\sin(2\theta) \]

We can express \( \sin(4\theta) \) as \( 2\sin(2\theta)\cos(2\theta) \):
\[ |A| = 8\cos^2(2\theta) + 1 - 3\sin(4\theta) \]

### Step 5: Find the maximum value
To find the maximum value of \( |A| \), we need to analyze the expression:
\[ |A| = 8\cos^2(2\theta) + 1 - 3\sin(4\theta) \]

The maximum value of \( -3\sin(4\theta) \) is \( 3 \) when \( \sin(4\theta) = -1 \). Thus:
\[ |A|_{\text{max}} = 8\cos^2(2\theta) + 1 + 3 \]
The maximum value of \( \cos^2(2\theta) \) is \( 1 \):
\[ |A|_{\text{max}} = 8 \cdot 1 + 1 + 3 = 12 \]

### Step 6: Determine \( k \) and calculate \( (k-3)^2 \)
From the calculations, we find \( k = 12 \). Therefore:
\[ (k - 3)^2 = (12 - 3)^2 = 9^2 = 81 \]

### Final Answer
Thus, the value of \( (k - 3)^2 \) is \( 81 \).