For what value of x , is the following matrix singular ?
`[(3-2x,x+1),(2,4)]`

Answer

To determine the value of \( x \) for which the given matrix

\[ A = \begin{pmatrix} 3 - 2x & x + 1 \\ 2 & 4 \end{pmatrix} \]

is singular, we need to find when the determinant of the matrix 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 \( A \):

- \( a = 3 - 2x \)
- \( b = x + 1 \)
- \( c = 2 \)
- \( d = 4 \)

Thus, the determinant can be calculated as follows:

\[ \text{det}(A) = (3 - 2x) \cdot 4 - (x + 1) \cdot 2 \]

### Step 2: Expand the determinant

Now, we expand the expression:

\[ \text{det}(A) = 4(3 - 2x) - 2(x + 1) \]

Calculating this gives:

\[ = 12 - 8x - 2x - 2 \]

### Step 3: Simplify the expression

Now, simplify the expression:

\[ = 12 - 2 - 8x - 2x \]
\[ = 10 - 10x \]

### Step 4: Set the determinant to zero

To find the value of \( x \) for which the matrix is singular, we set the determinant equal to zero:

\[ 10 - 10x = 0 \]

### Step 5: Solve for \( x \)

Now, solve for \( x \):

\[ 10 = 10x \]
\[ x = 1 \]

Thus, the value of \( x \) for which the matrix is singular is

\[ \boxed{1} \]

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