Question-85

Let \(\displaystyle A\) be a \(\displaystyle 3\times 4\) real matrix.

• The maximum rank of \(\displaystyle A\) is \(\displaystyle 3\)
• The minimum nullity of \(\displaystyle A\) is \(\displaystyle 1\)
• The minimum rank of \(\displaystyle A\) is \(\displaystyle 1\)
• The maximum nullity of \(\displaystyle A\) is \(\displaystyle 3\).

Hint

Can the matrix be zero?

Answer

• The maximum rank of \(\displaystyle A\) is \(\displaystyle 3\)
• The minimum nullity of \(\displaystyle A\) is \(\displaystyle 1\)
• The minimum rank of \(\displaystyle A\) is \(\displaystyle 1\)
• The maximum nullity of \(\displaystyle A\) is \(\displaystyle 3\).

Solution

If all three rows of \(\displaystyle A\) are linearly independent, \(\displaystyle \text{rank}( A) =3\). The rank cannot exceed the minimum dimension (row/column) of a matrix. Using rank nullity theorem, the minimum nullity is \(\displaystyle 1\). The minimum rank is \(\displaystyle 0\), if \(\displaystyle A\) happens to be a zero matrix. The maximum nullity of \(\displaystyle A\) is \(\displaystyle 4\), which again follows from rank nullity.