GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 10

Answer 1

Note that:

    Column 2 = Column1 * 2

    Column 4 = Column4 * 2/3

    Column 3 = (Column2 + Column 4) / 2

Thus there are only 2 linearly independent columns so rank is 2.

Answer 2

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

Given size of matrix is $3\times 4$ so rank should be $\rho\leq3$

Perform $R_2\rightarrow R_2-R_1,R_3\rightarrow R_3+R_1$ we get:

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

Perform $R_3\rightarrow R_3-R_2$;

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

here $R_3$ is completly zero so $\rho(A)\leq 2$

$\therefore \rho(A)=2$