Question

If \( A = \begin{bmatrix} 1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6 \end{bmatrix} \) and the rank of \( A \) is 2, then the value of \( x \) is equal to:

• A. \( 1 \)
• B. \( 0 \)
• C. \( -3 \)
• D. \( 3 \)

Hint:

To find the rank condition for a matrix, use row-reduction (Gaussian elimination) and force the required number of non-zero rows. Here, setting the third row to zero ensures rank 2 and helps find the unknown value.

Answer 1

The Correct Option is C

We are given the matrix: \[ A = \begin{bmatrix} 1 & 2 & x
4 & -1 & 7
2 & 4 & -6 \end{bmatrix} \] Step 1: Perform the row operation \( R_2 \leftarrow R_2 - 4R_1 \): \[ R_2 = [4, -1, 7] - 4 \cdot [1, 2, x] = [0, -9, 7 - 4x] \] Step 2: Perform the row operation \( R_3 \leftarrow R_3 - 2R_1 \): \[ R_3 = [2, 4, -6] - 2 \cdot [1, 2, x] = [0, 0, -6 - 2x] \] So the matrix becomes: \[ \begin{bmatrix} 1 & 2 & x \\ 0 & -9 & 7 - 4x \\ 0 & 0 & -6 - 2x \end{bmatrix} \] Step 3: For the matrix to have rank 2, one row must be a linear combination of others or a zero row. That is, third row must be zero: \[ -6 - 2x = 0 \Rightarrow x = -3 \] Therefore, the value of \( x \) is: \[ \boxed{-3} \]