Question

Determine the value of \( p \) such that the rank of matrix \( A \) is 2:
\[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 0 & 2 & p \\ 1 & 0 & p & 7 \end{bmatrix} \]

• A. \( \frac{8}{5} \)
• B. \( \frac{21}{5} \)
• C. \( \frac{20}{7} \)
• D. \( \frac{3}{4} \)

Hint:

To determine the rank of a matrix, find the determinant of its submatrices. If the determinant is zero, the rank is less than full.

Answer 1

The Correct Option is C

To find the value of \( p \) that makes the rank of matrix \( A \) equal to 2, we need to solve for the determinant of a \( 3 \times 3 \) submatrix and ensure that the determinant is zero for rank 2. After performing the necessary calculations, we find that \( p = \frac{20}{7} \).