The matrix `A=[{:(a,2),(2,4):}]` is singular if

Options

A. 1
B. -2
C. -1
D. 0

Answer

To determine when the matrix \( A = \begin{pmatrix} a & 2 \\ 2 & 4 \end{pmatrix} \) is singular, we need to find the conditions under which its determinant is equal to zero.

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

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 \( A = \begin{pmatrix} a & 2 \\ 2 & 4 \end{pmatrix} \):

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

Thus, the determinant is:

\[ \text{det}(A) = a \cdot 4 - 2 \cdot 2 \]

### Step 2: Simplify the determinant expression.

Calculating the above expression gives:

\[ \text{det}(A) = 4a - 4 \]

### Step 3: Set the determinant equal to zero for singularity.

For the matrix \( A \) to be singular, we set the determinant equal to zero:

\[ 4a - 4 = 0 \]

### Step 4: Solve for \( a \).

Now, we solve the equation:

\[ 4a - 4 = 0 \]

Adding 4 to both sides:

\[ 4a = 4 \]

Dividing both sides by 4:

\[ a = 1 \]

### Conclusion:

The matrix \( A \) is singular if \( a = 1 \).

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