Let [A = ( pmatrix 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 pmatrix )].

Question

Let $$A = \left( {\begin{pmatrix} 1 & 0 & 0  \
   2 & 1 & 0  \
   3 & 2 & 1  \ \end{pmatrix} } \right)$$. If ${u_1}$ and ${u_2}$ are column matrices such

that $$A{u_1} = \left( {\begin{pmatrix} 1  \
   0  \
   0  \ \end{pmatrix} } \right)$$ and $$A{u_2} = \left( {\begin{pmatrix} 0  \
   1  \
   0  \ \end{pmatrix} } \right),$$ then ${u_1} + {u_2}$ is equal to :

Accepted Answer

Let $$A{u_1} = \left( {\begin{pmatrix} 1  \
   0  \
   0  \

\end{pmatrix} } \right)\,\,\,\,\,\,A{u_2} = \left( {\begin{pmatrix} 0  \
   1  \
   0  \

\end{pmatrix} } \right)$$

Then, $$A{u_1} + A{u_2} = \left( {\begin{pmatrix} 1  \
   0  \
   0  \

\end{pmatrix} } \right) + \left( {\begin{pmatrix} 0  \
   1  \
   0  \

\end{pmatrix} } \right)$$

$$ \Rightarrow A\left( {{u_1} + {u_2}} \right) = \left( {\begin{pmatrix} 1  \
   1  \
   0  \

\end{pmatrix} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

Also, $$A = \left( {\begin{pmatrix} 1 & 0 & 0  \
   2 & 1 & 0  \
   3 & 2 & 1  \

\end{pmatrix} } \right)$$

$$ \Rightarrow \left| A \right| = 1\left( 1 \right) - 0\left( 2 \right) + 0\left( {4 - 3} \right) = 1$$

We know,

$${A^{ - 1}} = {1 \over {\left| A \right|}}\,adjA \Rightarrow {A^{ - 1}} = adj\left( A \right)$$

( as $$\left| A \right| = 1$$ )

Now, from equation $(1)$, we have

$${u_1} + {u_2} = {A^{ - 1}}\left( {\begin{pmatrix} 1  \
   1  \
   0  \

\end{pmatrix} } \right)$$

$$ = \left[ {\begin{pmatrix} 1 & 0 & 0  \
   { - 2} & 1 & 0  \
   1 & { - 2} & 1  \

\end{pmatrix} } \right]\left( {\begin{pmatrix} 1  \
   1  \
   0  \

\end{pmatrix} } \right)$$

$$ = \left[ {\begin{pmatrix} 1  \
   { - 1}  \
   { - 1}  \

\end{pmatrix} } \right]$$

Solutions · 1