Question

3 O o 12 3 Let A= o O o O 0 o o -4 O and the The dimension of Nul A is dimension of ColA is

Answer 1

$A=\begin{pmatrix}3&0&0&12&3\\ 0&0&1&4&-2\\ 0&0&0&0&0\\ 0&1&0&-4&1\end{pmatrix}$

$R_2\:\leftrightarrow \:R_4$

$=\begin{pmatrix}3&0&0&12&3\\ 0&1&0&-4&1\\ 0&0&0&0&0\\ 0&0&1&4&-2\end{pmatrix}$

R3 R4

= 3 0 0 12 3 0 1 0 -4 0 0 1 4 -2 0 0 0 0 0

$R_1\:\leftarrow \frac{1}{3}\cdot \:R_1$

$=\begin{pmatrix}{\color{Red} 1}&0&0&4&1\\ 0&{\color{Red} 1}&0&-4&1\\ 0&0&{\color{Red} 1}&4&-2\\ 0&0&0&0&0\end{pmatrix}$

there is no pivot entry at the fourth and fifth column

so the dimension of the null(A) is ${\color{Red} 2}$

.

and we have 3 pivot entry at the first second and third column

so the dimension of the col(A) is ${\color{Red} 3}$