Question

IfA is a$\mathbf{5}\mathbf{\times }\mathbf{6}$matrix of rank4, then the nullity ofAis1.

Short answer

The given statement is false. If A is a $5\times 6$matrix of rank 4, then the nullity of A is 2.

Step-by-step solution

Concept used

According to the rank-nullity theorem, for anymatrixA, the equation

(nullity ofA) + (rank ofA)= m

holds true. This theorem is also known as the fundamental theorem of linear algebra.

Given data

Given a matrixAwhich is a$5\times 6$ matrix of rank 4.

Applying the concept

In accordance with the rank-nullity theorem, the following equation holds true for the given matrixA

(nullity ofA) + (rank ofA)= m

From the given data, it can be briefed out that (rank ofA)=4and the value ofmturns out be6as the given matrix is of the order$5\times 6$.

Therefore, by substituting the values in the above equation, we get

(nullity of A)=6-4=2

Result

Therefore, it will be wrong to say that if A is a $5\times 6$matrix of rank 4, then the nullity of A is 1.