Question

If A and B are two matrices of same order and rank (A)=rank (B)=n ,
then rank (A+B)=n , for n>=1 .

Answer 1

Answer on Question #40788 – Math - Linear Algebra

If A and B are two matrices of same order and rank (A)=rank (B)=n, then rank (A+B)=n, for n>1.

Solution:

Let us show that this statement is false. Indeed if (A) = $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$, (B) = $\begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ we get rank(A)=rank(B)=2 (in this case n=2), but rank(A+B)=3. So, the statement above is false.

Answer: false.