If `A=[[2,a,-3] , [0,2,5] , [1,1,3]]` then,find the value of a for which `A^(-1)` exists.

Options

A. `a=2`
B. `a!=2`
C. `a!=-2`
D. none of these

Explanation

To find the value of \( a \) for which the inverse of the matrix \( A \) exists, we need to ensure that the determinant of \( A \) is not equal to zero. The matrix \( A \) is given as:

\[ A = \begin{bmatrix} 2 & a & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{bmatrix} \]

### Step 1: Calculate the Determinant of Matrix A

We will calculate the determinant of \( A \) using the formula for the determinant of a \( 3 \times 3 \) matrix:

\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

Substituting the values from matrix \( A \):

\[ \text{det}(A) = 2 \cdot (2 \cdot 3 - 5 \cdot 1) - a \cdot (0 \cdot 3 - 5 \cdot 1) - 3 \cdot (0 \cdot 1 - 2 \cdot 1) \]

Calculating each term:

1. \( 2 \cdot (6 - 5) = 2 \cdot 1 = 2 \)
2. \( -a \cdot (0 - 5) = 5a \)
3. \( -3 \cdot (0 - 2) = 6 \)

Putting it all together:

\[ \text{det}(A) = 2 + 5a + 6 = 5a + 8 \]

### Step 2: Set the Determinant Not Equal to Zero

For the inverse of \( A \) to exist, we need:

\[ \text{det}(A) \neq 0 \]

Thus, we set up the inequality:

\[ 5a + 8 \neq 0 \]

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

Now we solve for \( a \):

\[ 5a \neq -8 \]

Dividing both sides by 5:

\[ a \neq -\frac{8}{5} \]

### Conclusion

The value of \( a \) for which \( A^{-1} \) exists is any real number except \( -\frac{8}{5} \).

### Final Answer

\[ a \in \mathbb{R} \setminus \left\{-\frac{8}{5}\right\} \]