Find the value of 'k' for which the following matrices are invertible ?
(i) `|{:(6,k),(-2,1):}|`
(ii) `|{:(0,k,3),(1,-2,2),(4,3,-1):}|`

Answer

To determine the values of \( k \) for which the given matrices are invertible, we need to find the determinants of the matrices and set them to be non-zero.

### (i) Matrix:
\[ \begin{pmatrix} 6 & k \\ -2 & 1 \end{pmatrix} \]

**Step 1: Calculate the determinant of the matrix.**

The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{det} = ad - bc \]

For our matrix:
- \( a = 6 \)
- \( b = k \)
- \( c = -2 \)
- \( d = 1 \)

So, the determinant is:
\[ \text{det} = (6)(1) - (k)(-2) = 6 + 2k \]

**Step 2: Set the determinant not equal to zero.**

For the matrix to be invertible:
\[ 6 + 2k \neq 0 \]

**Step 3: Solve for \( k \).**

Rearranging the equation:
\[ 2k \neq -6 \]
\[ k \neq -3 \]

### Conclusion for (i):
The matrix is invertible for all values of \( k \) except \( k = -3 \).

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### (ii) Matrix:
\[ \begin{pmatrix} 0 & k & 3 \\ 1 & -2 & 2 \\ 4 & 3 & -1 \end{pmatrix} \]

**Step 1: Calculate the determinant of the matrix.**

Using the determinant formula for a \( 3 \times 3 \) matrix:
\[ \text{det} = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

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

Calculating the determinant:
\[ \text{det} = 0 \cdot ((-2)(-1) - (2)(3)) - k \cdot (1 \cdot (-1) - (2)(4)) + 3 \cdot (1 \cdot 3 - (-2)(4)) \]

This simplifies to:
\[ = 0 - k \cdot (-1 - 8) + 3 \cdot (3 + 8) \]
\[ = 0 + 9k + 33 \]
\[ = 9k + 33 \]

**Step 2: Set the determinant not equal to zero.**

For the matrix to be invertible:
\[ 9k + 33 \neq 0 \]

**Step 3: Solve for \( k \).**

Rearranging the equation:
\[ 9k \neq -33 \]
\[ k \neq -\frac{33}{9} \]
\[ k \neq -\frac{11}{3} \]

### Conclusion for (ii):
The matrix is invertible for all values of \( k \) except \( k = -\frac{11}{3} \).

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### Final Answer:
1. For the first matrix, \( k \neq -3 \).
2. For the second matrix, \( k \neq -\frac{11}{3} \).