For what value of 'x' the matrix `[{:(2-x,3),(-5,-1):}]` is not invertible ?

Answer

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

\[ \begin{pmatrix} 2 - x & 3 \\ -5 & -1 \end{pmatrix} \]

is not invertible, we need to find the condition under which the determinant of the matrix is equal to zero.

### Step 1: Calculate the Determinant

The determinant of a 2x2 matrix

\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

is given by the formula:

\[ \text{det} = ad - bc \]

For our matrix, we have:

- \( a = 2 - x \)
- \( b = 3 \)
- \( c = -5 \)
- \( d = -1 \)

Substituting these values into the determinant formula, we get:

\[ \text{det} = (2 - x)(-1) - (3)(-5) \]

### Step 2: Simplify the Determinant

Now, we simplify the expression:

\[ \text{det} = -(2 - x) + 15 \]

This simplifies to:

\[ \text{det} = -2 + x + 15 = x + 13 \]

### Step 3: Set the Determinant to Zero

To find the value of \( x \) for which the matrix is not invertible, we set the determinant equal to zero:

\[ x + 13 = 0 \]

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

Now, we solve for \( x \):

\[ x = -13 \]

### Conclusion

Thus, the matrix is not invertible when \( x = -13 \).

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