For what value(s) of k the matrix `[(4,k),(2,1)]` has no inverse?

Options

A. 2
B. 3
C. 4
D. 5

Answer

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

\[ A = \begin{pmatrix} 4 & k \\ 2 & 1 \end{pmatrix} \]

has no inverse, we need to find when the matrix is singular. A matrix is singular when its determinant is equal to zero.

### 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 the formula:

\[ \text{det}(A) = ad - bc \]

For our matrix \( A \):

- \( a = 4 \)
- \( b = k \)
- \( c = 2 \)
- \( d = 1 \)

Thus, the determinant can be calculated as follows:

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

### Step 2: Set the determinant equal to zero

To find when the matrix is singular, we set the determinant equal to zero:

\[ 4 - 2k = 0 \]

### Step 3: Solve for \( k \)

Now, we solve the equation for \( k \):

\[ 4 - 2k = 0 \]

Rearranging gives:

\[ 2k = 4 \]

Dividing both sides by 2:

\[ k = \frac{4}{2} = 2 \]

### Conclusion

The value of \( k \) for which the matrix has no inverse is:

\[ k = 2 \]