The matrix `[(2,-1,3),(lamda,0,7),(-1,1,4)]` is not invertible for

Options

A. `lamda=-1`
B. `lamda=1`
C. `lamda=0`
D. `lamda in R-{1}`

Answer

To determine the values of \(\lambda\) for which the matrix

\[ \begin{pmatrix} 2 & -1 & 3 \\ \lambda & 0 & 7 \\ -1 & 1 & 4 \end{pmatrix} \]

is not invertible, we need to find when the determinant of the matrix is equal to zero. A matrix is not invertible (or singular) if its determinant is zero.

### Step 1: Calculate the Determinant

The determinant of a \(3 \times 3\) matrix

\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

is calculated using the formula:

\[ \text{det} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix, we have:

- \(a = 2\), \(b = -1\), \(c = 3\)
- \(d = \lambda\), \(e = 0\), \(f = 7\)
- \(g = -1\), \(h = 1\), \(i = 4\)

Plugging these values into the determinant formula:

\[ \text{det} = 2(0 \cdot 4 - 7 \cdot 1) - (-1)(\lambda \cdot 4 - 7 \cdot -1) + 3(\lambda \cdot 1 - 0 \cdot -1) \]

### Step 2: Simplify the Determinant Expression

Calculating each term:

1. \(2(0 - 7) = 2 \cdot -7 = -14\)
2. \(-(-1)(\lambda \cdot 4 + 7) = \lambda \cdot 4 + 7\)
3. \(3(\lambda - 0) = 3\lambda\)

Putting it all together:

\[ \text{det} = -14 + (4\lambda + 7) + 3\lambda \]

Combine like terms:

\[ \text{det} = -14 + 7 + 4\lambda + 3\lambda = -7 + 7\lambda \]

### Step 3: Set the Determinant to Zero

To find when the matrix is not invertible, set the determinant to zero:

\[ -7 + 7\lambda = 0 \]

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

Rearranging the equation:

\[ 7\lambda = 7 \]

Dividing both sides by 7:

\[ \lambda = 1 \]

### Conclusion

The matrix is not invertible when \(\lambda = 1\).