The rank of the matrix `[[-1,2,5],[2,-4,a-4],[1,-2,a+1]]` is (where a = - 6)

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To find the rank of the matrix

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

where \( a = -6 \), we will follow these steps:

### Step 1: Substitute the value of \( a \)

First, we substitute \( a = -6 \) into the matrix:

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

This simplifies to:

\[ A = \begin{bmatrix} -1 & 2 & 5 \\ 2 & -4 & -10 \\ 1 & -2 & -5 \end{bmatrix} \]

### Step 2: Perform row operations to simplify the matrix

We will now perform row operations to bring the matrix to row echelon form.

1. **Row Operation 1**: Replace \( R_2 \) with \( R_2 + 2R_1 \):

\[ R_2 = R_2 + 2R_1 \Rightarrow R_2 = \begin{bmatrix} 2 & -4 & -10 \end{bmatrix} + 2 \cdot \begin{bmatrix} -1 & 2 & 5 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix} \]

Now the matrix looks like:

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

2. **Row Operation 2**: Replace \( R_3 \) with \( R_3 + R_1 \):

\[ R_3 = R_3 + R_1 \Rightarrow R_3 = \begin{bmatrix} 1 & -2 & -5 \end{bmatrix} + \begin{bmatrix} -1 & 2 & 5 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix} \]

Now the matrix is:

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

### Step 3: Determine the rank

The rank of a matrix is defined as the maximum number of linearly independent rows. In the final matrix, we can see that there is only **one non-zero row**.

Thus, the rank of the matrix \( A \) is:

\[ \text{Rank}(A) = 1 \]

### Final Answer

The rank of the matrix is **1**.

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