Question

Consider two n x m matrices A and B. What can you say about the relationship among the quantities rank(A), rank(B), rank(A+B).

Short answer

If we have two n x m matrices A and B then

$\mathrm{rank}(\mathrm{A}+\mathrm{B})\le \mathrm{rank}\left(\mathrm{A}\right)+\mathrm{rank}\left(\mathrm{B}\right).$

Step-by-step solution

  Mentioning concept

Since we know that

Dim (Im(A)) = rank(A) and dim (Im(B)) = rank(B)

So, dim (Im(A+B)) = rank(A+B)

  Finding the rank of (A+B)

Let$A=\left[{\overrightarrow{v}}_1{\overrightarrow{v}}_2...{\overrightarrow{v}}_m\right]\mathrm{and}B=[w{⃗}_{1}w{⃗}_2...w{⃗}_m]$

Where$v{⃗}_{i}andw{⃗}_i$ are column vectors of A and B respectively.

Then,

role="math" localid="1659355348377" $$\begin{gathered} \mathrm{rank}\left(\mathrm{A}\right)=\dim \left(\mathrm{span}\right(\mathrm{v}{⃗}_1,\mathrm{v}{⃗}_2,...,\mathrm{v}{⃗}_{\mathrm{m}}\left)\right) \\ \mathrm{and} \\ \mathrm{rank}\left(\mathrm{B}\right)=\dim \left(\mathrm{span}\right(\mathrm{w}{⃗}_1,\mathrm{w}{⃗}_2,...\mathrm{w}{⃗}_{\mathrm{m}}\left)\right) \end{gathered}$$

$\Rightarrow rank(A+B)=dim\left(span\right(v{⃗}_1+w{⃗}_1,v{⃗}_2+w{⃗}_2,...v{⃗}_m+w{⃗}_m\left)\right)$

Finding the dimension of image of (A+B)

Let us claim,

$span(v{⃗}_1+w{⃗}_1,\overrightarrow{v}{⃗}_2+w{⃗}_2,...,v{⃗}_m+w{⃗}_m)\subset span(v{⃗}_1,v{⃗}_2,...,v{⃗}_m)+span(w{⃗}_1,w{⃗}_2,...,w{⃗}_m)$

Then any vector can be written as

$$\begin{gathered} x⃗=a_1(v{⃗}_1+w{⃗}_1)+a_2(v{⃗}_2+w{⃗}_2)+...+a_m(v{⃗}_m+w{⃗}_m) \\ \Rightarrow x⃗=(a_{1}v{⃗}_1+a_{2}v{⃗}_2+...+a_{m}v{⃗}_m)+(a_{1}w{⃗}_1+a_{2}w{⃗}_2+...+a_{m}w{⃗}_m) \\ \Rightarrow x⃗\in span(v{⃗}_1,v{⃗}_2,...,v{⃗}_m)+span(w{⃗}_1,w{⃗}_2,...,w{⃗}_m) \end{gathered}$$

for some scalars $a{⃗}_1,a{⃗}_2,...,a_m$.

$$\begin{gathered} \Rightarrow span(v{⃗}_1+w{⃗}_1,v{⃗}_2+w{⃗}_2,...,v{⃗}_m+w{⃗}_m)\subseteq span(v{⃗}_1,v{⃗}_2,...,v{⃗}_m)+span(w{⃗}_1,w{⃗}_2,...,w{⃗}_m) \\ \Rightarrow Im(A+B)\subseteq Im\left(A\right)+Im\left(B\right) \\ \Rightarrow dim\left(Im\right(A+B\left)\right)\le dim\left(Im\right(A\left)\right)+dim\left(Im\right(B\left)\right) \\ \Rightarrow rank(A+B)\le rank\left(A\right)+rank\left(B\right) \end{gathered}$$

  Final Answer

If we have two n x m matrices A and B then

$$\begin{gathered} \Rightarrow Im(A+B)\subseteq Im\left(A\right)+Im\left(B\right) \\ \Rightarrow dim\left(Im\right(A+B\left)\right)\le dim\left(Im\right(A\left)\right)+dim\left(Im\right(B\left)\right) \\ \Rightarrow rank(A+B)\le rank\left(A\right)+rank\left(B\right) \end{gathered}$$