Question-63
Let \(\displaystyle A,B,C\) be three matrices such that \(\displaystyle A=BC\), with \(\displaystyle B\) having dimensions \(\displaystyle 4\times 3\) and \(\displaystyle C\) having dimensions \(\displaystyle 3\times 4\). Which of the following statements are true?
The rank of both \(\displaystyle B\) and \(\displaystyle C\) is at most \(\displaystyle 3\). Therefore, the rank of \(\displaystyle A=BC\) is at most \(\displaystyle 3\). From this it follows that the nullity of \(\displaystyle A\) is at least \(\displaystyle 1\). This rests on the following fact:
If \(\displaystyle A=BC\), the rank of \(\displaystyle A\) is at most \(\displaystyle \min\left(\text{rank}( B) ,\text{rank}( C)\right)\).
Let \(\displaystyle B\) be of shape \(\displaystyle m\times p\) and \(\displaystyle C\) be of shape \(\displaystyle p\times n\). We can express \(\displaystyle A\) as
\[ \begin{equation*} A=\begin{bmatrix} | & & |\\ Bc_{1} & \cdots & Bc_{n}\\ | & & | \end{bmatrix} \end{equation*} \]
\(\displaystyle Bc_{i}\) is a linear combination of the columns of \(\displaystyle B\). The span of the columns of \(\displaystyle A\), which is the column space of \(\displaystyle A\), is therefore a subset of the column space of \(\displaystyle B\). Thus, we have \(\displaystyle \text{rank}( A) \leqslant \text{rank}( B)\).
We also have:
\[ \begin{equation*} A=\begin{bmatrix} - & b_{1}^{T} C & -\\ & \vdots & \\ - & b_{m}^{T} C & - \end{bmatrix} \end{equation*} \]
\(\displaystyle b_{i}^{T} C\) is a linear combination of the rows of \(\displaystyle C\). The span of the rows of \(\displaystyle A\), which is the rowspace of \(\displaystyle A\), is therefore a subset of the row space of \(\displaystyle C\). Thus, we have \(\displaystyle \text{rank}( A) \leqslant \text{rank}( C)\). Here, we have used the fact that column rank equals the row rank.
Combining the two results, we get:
\[ \begin{equation*} \text{rank}( A) \leqslant \min\left(\text{rank}( B) ,\text{rank}( C)\right) \end{equation*} \]