## Matrix problem (proofs)

$Let A=
\begin{bmatrix}
a1&a2&a3 \\
\ b1&b2&b3 \\
\ c1&c2&c3\\
\end{bmatrix}
$

$B=
\begin{bmatrix}
a1 & a2 \\
\ b1 & b2 \\
\ c1&c2\\
\end{bmatrix}$

Suppose B is nonsingular. $Bz=
\begin{bmatrix}
a3 \\
\ b3 \\
\end{bmatrix}$

Let x be a 3-vector. Show that

$\begin{bmatrix}
a1 & a2 & a3 \\
\ b1 & b2 & b3 \\
\end{bmatrix}\
$
x
=
$\begin{bmatrix}
0\\
\ 0 \\
\end{bmatrix}$

if and only if there is a scalar $\lambda$ such that
$x=
\lambda
\begin{bmatrix}
z1\\
\ z2 \\
-1\\
\end{bmatrix}$

b)Show that A is singular if and only if c1z1 + c2z2 = c3