Select and write the most appropriate answer from the given alternatives in each of the following :
If the matrix `{:((6,-5,1),(4,2,-1),(14,-1,k)):}` has no inverse, then

Options

A. k = 1
B. `k = - 1`
C. ` k = 0`
D. ` k = 2`

Answer

To determine the value of \( k \) for which the matrix

\[ A = \begin{pmatrix} 6 & -5 & 1 \\ 4 & 2 & -1 \\ 14 & -1 & k \end{pmatrix} \]

has no inverse, we need to find the determinant of the matrix and set it equal to zero. A matrix has no inverse if its determinant is zero.

### Step 1: Calculate the determinant of the matrix \( A \)

The determinant of a \( 3 \times 3 \) matrix

\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

is given by the formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A \):

- \( a = 6, b = -5, c = 1 \)
- \( d = 4, e = 2, f = -1 \)
- \( g = 14, h = -1, i = k \)

Substituting these values into the determinant formula:

\[ \text{det}(A) = 6(2k - (-1)(-1)) - (-5)(4k - (-1)(14)) + 1(4(-1) - 2(14)) \]

### Step 2: Simplify the determinant expression

Calculating each term:

1. First term:
\[ 6(2k - 1) = 12k - 6 \]

2. Second term:
\[ -5(4k - 14) = -20k + 70 \]

3. Third term:
\[ 1(-4 - 28) = -32 \]

Putting it all together:

\[ \text{det}(A) = (12k - 6) + (-20k + 70) - 32 \]

Combine like terms:

\[ \text{det}(A) = 12k - 20k + 70 - 6 - 32 = -8k + 32 \]

### Step 3: Set the determinant equal to zero

For the matrix to have no inverse, we set the determinant to zero:

\[ -8k + 32 = 0 \]

### Step 4: Solve for \( k \)

Rearranging gives:

\[ -8k = -32 \]

Dividing both sides by -8:

\[ k = 4 \]

### Conclusion

The value of \( k \) for which the matrix has no inverse is \( k = 4 \).