Question-2

What is the maximum rank that a \(4 \times 8\)​ matrix can have?

Hint

The rank is the dimension of the row space. It is also equal to the dimension of the column space.

Answer

\(4\)

Solution

The row rank is equal to the column rank for any matrix. The row rank is the dimension of the row space. This can be at most \(n\). The column rank is the dimension of the column space. This can be at most \(m\). Therefore, the rank of a \(m \times n\) matrix is at most \(\min(m, n)\)​.