# Show that block matrix has full row rank ii and only if $A$ has full row rank.

I've read a text, and they say the following lemma,

Lemma: For given matrices $$A,B$$ , the matrix $$\begin{pmatrix} A & B\\ 0^T & v^T\\ \end{pmatrix}$$ has full row rank for all nonzero $$v$$ iif $$A$$ has full row rank.

where the matrix, not needed to be squared. can give me som hint, thanks!

• Hint: A matrix $M$ has full row-rank if and only if the system $M^Tx = 0$ (or equivalently $xM = 0$ for a row-vector $x$) has only the trivial solution May 28 '21 at 19:47

If $$A$$ does not have full row rank, then there exists a nonzero $$u$$ such that $$u^\top A = 0$$. Try to come up with a nonzero vector $$v$$ and scalar $$c$$ such that $$\begin{pmatrix} u^\top & c \end{pmatrix} \begin{pmatrix} A & B \\ 0^\top & v^\top \end{pmatrix} = 0.$$ (It will depend on what $$u^\top B$$ is.)

If there exists nonzero $$v$$ such that $$M := \begin{pmatrix} A & B \\ 0^\top & v^\top \end{pmatrix}$$ does not have full row rank, then there exists a vector $$u$$ and scalar $$c$$ such that $$\begin{pmatrix}u \\ c \end{pmatrix}$$ is nonzero and $$\begin{pmatrix} u^\top & c \end{pmatrix} \begin{pmatrix} A & B \\ 0^\top & v^\top \end{pmatrix} = 0.$$

Show that the above conditions imply that $$u$$ is nonzero and that $$u^\top A = 0$$.

call your overall matrix $$C$$ which has $$m$$ rows .

1.) $$A \text{ is full row rank }\implies C \text{ is full row rank}$$
since $$A$$ has full row rank, it is surjective and has a right inverse. Thus there is some (block) matrix $$W$$ such that
$$CW = \begin{pmatrix} A & B\\ \mathbf 0^T & \mathbf v^T\\ \end{pmatrix}W = \begin{pmatrix} I_{m-1} & *\\ \mathbf 0^T & \mathbf v^T\\ \end{pmatrix}$$

$$m = \text{rank}\big(CW\big)\leq \text{rank}\big(C\big)\leq m$$

2.) $$C \text{ is full row rank} \implies A\text{ is full row rank }$$
$$C$$ is sujective so it has a right inverse $$W'$$
$$CW' = I_m$$
This implies that for all non-zero $$\mathbf x':=\begin{pmatrix}\mathbf x \\ 0 \end{pmatrix}$$
$$(\mathbf x')^TC \neq\mathbf 0^T\implies \mathbf x^TA\neq \mathbf 0^T$$
and hence all rows of $$A$$ are linearly independent.