If for a matrix `A`, `absA=6` and `adj A={:[(1,-2,4),(4,1,1),(-1,k,0)]:}`, then `k` is equal to

Explanation

To find the value of \( k \) in the given matrix \( \text{adj } A \), we can use the property of the adjugate matrix and the determinant. Here are the steps to solve the problem:

### Step 1: Understand the relationship between the determinant and the adjugate
The relationship between a matrix \( A \), its adjugate \( \text{adj } A \), and its determinant \( \text{det } A \) is given by:
\[ A \cdot \text{adj } A = \text{det } A \cdot I \]
where \( I \) is the identity matrix.

### Step 2: Use the determinant of \( A \)
Given that \( |\text{det } A| = 6 \), we can say:
\[ \text{det } A = 6 \]

### Step 3: Write down the adjugate matrix
The adjugate matrix \( \text{adj } A \) is given as:
\[ \text{adj } A = \begin{pmatrix} 1 & -2 & 4 \\ 4 & 1 & 1 \\ -1 & k & 0 \end{pmatrix} \]

### Step 4: Calculate the determinant of the adjugate matrix
The determinant of a \( 3 \times 3 \) matrix can be calculated using the formula:
\[ \text{det }(B) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our adjugate matrix:
\[ B = \begin{pmatrix} 1 & -2 & 4 \\ 4 & 1 & 1 \\ -1 & k & 0 \end{pmatrix} \]
We can denote:
- \( a = 1, b = -2, c = 4 \)
- \( d = 4, e = 1, f = 1 \)
- \( g = -1, h = k, i = 0 \)

Now, substituting these values into the determinant formula:
\[ \text{det }(B) = 1(1 \cdot 0 - 1 \cdot k) - (-2)(4 \cdot 0 - 1 \cdot -1) + 4(4 \cdot k - 1 \cdot -1) \]
This simplifies to:
\[ \text{det }(B) = 1(0 - k) + 2(0 + 1) + 4(4k + 1) \]
\[ = -k + 2 + 16k + 4 \]
\[ = 15k + 6 \]

### Step 5: Relate the determinant of the adjugate to the determinant of \( A \)
From the properties of determinants, we know:
\[ \text{det }(\text{adj } A) = (\text{det } A)^{n-1} \]
For a \( 3 \times 3 \) matrix, \( n = 3 \):
\[ \text{det }(\text{adj } A) = (\text{det } A)^{3-1} = (\text{det } A)^2 = 6^2 = 36 \]

### Step 6: Set the determinant of the adjugate equal to 36
Now we equate the determinant we calculated to 36:
\[ 15k + 6 = 36 \]

### Step 7: Solve for \( k \)
Subtract 6 from both sides:
\[ 15k = 30 \]
Now divide by 15:
\[ k = 2 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{2} \]