The matrix `[[1, a, 2], [1, 2, 5], [2, 1, 1]]` is not invertible, if a=

Options

A. `-1`
B. `0`
C. `1`
D. `2`

Answer

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

\[ \begin{bmatrix} 1 & a & 2 \\ 1 & 2 & 5 \\ 2 & 1 & 1 \end{bmatrix} \]

is not invertible, we need to find when the determinant of the matrix is equal to zero.

### Step 1: Write down the determinant of the matrix

The determinant of a 3x3 matrix

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

can be calculated using the formula:

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

For our matrix, we have:

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

### Step 2: Calculate each term in the determinant

1. Calculate \( 2 \cdot 1 - 5 \cdot 1 \):
\[ 2 - 5 = -3 \]

2. Calculate \( 1 \cdot 1 - 5 \cdot 2 \):
\[ 1 - 10 = -9 \]

3. Calculate \( 1 \cdot 1 - 2 \cdot 2 \):
\[ 1 - 4 = -3 \]

### Step 3: Substitute back into the determinant formula

Now substituting these values back into the determinant expression:

\[ \text{det} = 1 \cdot (-3) - a \cdot (-9) + 2 \cdot (-3) \]

This simplifies to:

\[ \text{det} = -3 + 9a - 6 \]

### Step 4: Combine like terms

Combining the constant terms:

\[ \text{det} = 9a - 9 \]

### Step 5: Set the determinant equal to zero

For the matrix to be non-invertible, we set the determinant to zero:

\[ 9a - 9 = 0 \]

### Step 6: Solve for \( a \)

Solving for \( a \):

\[ 9a = 9 \\ a = 1 \]

### Conclusion

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

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