If `A=|{:(2,lambda,-3),(0,2,5),(1,1,3):}|`. Then `A^(-1)` exist if

Options

A. `lambda=2`
B. `lambda ne 2`
C. `lambda ne -2`
D. None of these

Answer

To determine the conditions under which the inverse of the matrix \( A \) exists, we need to calculate the determinant of the matrix \( A \) and find the values of \( \lambda \) for which this determinant is not equal to zero.

Given the matrix:
\[ A = \begin{pmatrix} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{pmatrix} \]

### Step 1: Calculate the Determinant of Matrix \( A \)

The determinant of a \( 3 \times 3 \) matrix can be calculated using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is represented as:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix \( A \):
- \( a = 2 \), \( b = \lambda \), \( c = -3 \)
- \( d = 0 \), \( e = 2 \), \( f = 5 \)
- \( g = 1 \), \( h = 1 \), \( i = 3 \)

Using the determinant formula:
\[ \text{det}(A) = 2 \cdot (2 \cdot 3 - 5 \cdot 1) - \lambda \cdot (0 \cdot 3 - 5 \cdot 1) + (-3) \cdot (0 \cdot 1 - 2 \cdot 1) \]

### Step 2: Simplify the Determinant Expression

Calculating each term:
- First term: \( 2 \cdot (6 - 5) = 2 \cdot 1 = 2 \)
- Second term: \( -\lambda \cdot (0 - 5) = 5\lambda \)
- Third term: \( -3 \cdot (0 - 2) = 6 \)

Putting it all together:
\[ \text{det}(A) = 2 + 5\lambda + 6 \]
\[ \text{det}(A) = 5\lambda + 8 \]

### Step 3: Set the Determinant Not Equal to Zero

For the inverse of \( A \) to exist, we need:
\[ \text{det}(A) \neq 0 \]
This gives us the condition:
\[ 5\lambda + 8 \neq 0 \]

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

Solving the inequality:
\[ 5\lambda \neq -8 \]
\[ \lambda \neq -\frac{8}{5} \]

### Conclusion

The inverse of the matrix \( A \) exists if:
\[ \lambda \neq -\frac{8}{5} \]