The rank of the matrix `[{:(-1,2,5),(2,-4,a-4),(1,-2,a+1):}]` is

Options

A. 1 if a = 6
B. 2 if a = -1
C. 3 if a = 2
D. 1 if a = -6

Answer

To find the rank of the matrix \[ A = \begin{pmatrix} -1 & 2 & 5 \\ 2 & -4 & a-4 \\ 1 & -2 & a+1 \end{pmatrix} \], we will perform row operations to simplify it and determine the rank based on the number of non-zero rows.

### Step 1: Row Operation
We will perform the operation \( R_3 \leftarrow R_3 + R_1 \) to simplify the third row.

\[ R_3 = (1, -2, a+1) + (-1, 2, 5) = (0, 0, a + 6) \]

Now the matrix looks like this:

\[ A = \begin{pmatrix} -1 & 2 & 5 \\ 2 & -4 & a-4 \\ 0 & 0 & a+6 \end{pmatrix} \]

### Step 2: Further Row Operation
Next, we will perform the operation \( R_2 \leftarrow R_2 + 2R_1 \) to simplify the second row.

\[ R_2 = (2, -4, a-4) + 2(-1, 2, 5) = (0, 0, a + 6) \]

Now the matrix becomes:

\[ A = \begin{pmatrix} -1 & 2 & 5 \\ 0 & 0 & a + 6 \\ 0 & 0 & a + 6 \end{pmatrix} \]

### Step 3: Determine the Rank
Now we analyze the matrix:

1. The first row \((-1, 2, 5)\) is non-zero.
2. The second and third rows are the same, both being \((0, 0, a + 6)\).

The rank of a matrix is determined by the number of linearly independent rows.

- If \( a + 6 \neq 0 \), then the second row is non-zero, and the rank is 2.
- If \( a + 6 = 0 \) (i.e., \( a = -6 \)), then both the second and third rows become zero, and the rank is 1.

### Conclusion
Thus, the rank of the matrix depends on the value of \( a \):

- If \( a = -6 \), the rank is 1.
- If \( a \neq -6 \), the rank is 2.

### Final Answer
The rank of the matrix is:
- 1 if \( a = -6 \)
- 2 if \( a \neq -6 \)