Question

A is a \( m \times n \) matrix where \( m>7 \) and \( n>8 \). If all the minors of the 7th order of A vanish and there is a 6th order non-zero minor existing for A, then rank of A is

• A. \( \leq 5 \)
• B. 5
• C. 6
• D. \( \leq 6 \)

Hint:

If \( r \)-th order minor is non-zero and all \( (r+1) \)-th order minors are zero, then rank = \( r \).

Answer 1

The Correct Option is C

Explanation:

To determine the rank of a given matrix \(A\), we need to consider its minors. A minor is the determinant of some smaller square matrix, cut down from the main matrix.

Given that all the minors of the 7th order of the matrix \(A\) vanish, it implies that any 7x7 submatrix of \(A\) has a determinant of zero. This means the rank of the matrix \(A\) cannot be 7 or greater because there is no non-zero determinant of a 7th order.

Additionally, we have information that there is a non-zero minor of 6th order existing, which means there is at least one submatrix of order 6 with a non-zero determinant. This tells us that the matrix has at least one set of linearly independent rows and columns of order 6.

Therefore, the rank of matrix \(A\) must be 6.

Thus, the rank of matrix \(A\) is 6.