Let A be a [3 3] real matrix such that [A([gathered 1 0 1 gathered])=2([gathered 1 0 1 gat…

Question

Let $A$ be a $$3 \times 3$$ real matrix such that

$$A\left(\[\begin{gathered} 1 \ 0 \ 1 \end{gathered}$$\right)=2\left($$\begin{gathered} 1 \ 0 \ 1 \end{gathered}$$\right), A\left($$\begin{gathered} -1 \ 0 \ 1 \end{gathered}$$\right)=4\left($$\begin{gathered} -1 \ 0 \ 1 \end{gathered}$$\right), A\left($$\begin{gathered} 0 \ 1 \ 0 \end{gathered}$$\right)=2\left($$\begin{gathered} 0 \ 1 \ 0 \end{gathered}$$\right) \text {. }\]

Then, the system $$(A-3 I)\left(\[\begin{gathered} x \ y \ z \end{gathered}$$\right)=\left($$\begin{gathered} 1 \ 2 \ 3 \end{gathered}$$\right)\] has :

Accepted Answer

$$\text { Let } A=\left[\[\begin{array}{lll} x_1 & y_1 & z_1 \ x_2 & y_2 & z_2 \ x_3 & y_3 & z_3 \end{array}$$\right]\]

$$\text { Given } A\left[\[\begin{gathered} 1 \ 0 \ 1 \end{gathered}$$\right]=\left[$$\begin{gathered} 2 \ 0 \ 2 \end{gathered}$$\right] \quad \text{ ..... (1)}\]

$$\therefore\left[\[\begin{gathered} \mathrm{x}_1+\mathrm{z}_1 \ \mathrm{x}_2+\mathrm{z}_2 \ \mathrm{x}_3+\mathrm{z}_3 \end{gathered}$$\right]=\left[$$\begin{gathered} 2 \ 0 \ 2 \end{gathered}$$\right]\]

$$\begin{aligned}
\therefore \mathrm{x}_1+\mathrm{z}_1 & =2 \quad \text{.... (2)}\
\mathrm{x}_2+\mathrm{z}_2 & =0 \quad \text{.... (3)}\
\mathrm{x}_3+\mathrm{z}_3 & =0 \quad \text{.... (4)}
\end{aligned}$$

$$\begin{aligned}
& \text { Given } A\left[\[\begin{gathered} -1 \ 0 \ 1 \end{gathered}\end{aligned}$$\right]=\left[$$\begin{gathered} -4 \ 0 \ 4 \end{gathered}$$\right] \
\& \therefore\left[$$\begin{gathered} -\mathrm{x}_1+\mathrm{z}_1 \ -\mathrm{x}_2+\mathrm{z}_2 \ -\mathrm{x}_3+\mathrm{z}_3 \end{gathered}$$\right]=\left[$$\begin{gathered} 4 \ 0 \ 4 \end{gathered}$$\right]
\]

$$\[\begin{gathered} \Rightarrow-\mathrm{x}_1+\mathrm{z}_1=-4 \quad \text{... (5)} \ -\mathrm{x}_2+\mathrm{x}_2=0 \quad \text{... (6)} \ -\mathrm{x}_3+\mathrm{z}_3=4 \end{gathered}$$\]

$$\begin{aligned}
& \text { Given } A\left[\[\begin{gathered} 0 \ 1 \ 0 \end{gathered}\end{aligned}$$\right]=\left[$$\begin{gathered} 0 \ 2 \ 0 \end{gathered}$$\right] \
\& \therefore\left[$$\begin{gathered} \mathrm{y}_1 \ \mathrm{y}_2 \ \mathrm{y}_3 \end{gathered}$$\right]=\left[$$\begin{gathered} 0 \ 2 \ 0 \end{gathered}$$\right]
\]

$$\begin{aligned}
& \therefore \mathrm{y}_1=0, \mathrm{y}_2=2, \mathrm{y}_3=0 \
& \therefore \text { from }(2),(3),(4),(5),(6) \text { and }(7) \
& \mathrm{x}_1=3 \mathrm{x}, \mathrm{x}_2=0, \mathrm{x}_3=-1 \
& \mathrm{y}_1=0, \mathrm{y}_2=2, \mathrm{y}_3=0 \
& \mathrm{z}_1=-1, \mathrm{z}_2=0, \mathrm{z}_3=3
\end{aligned}$$

$$\begin{gathered}
\therefore A=\left[\[\begin{array}{ccc} 3 & 0 & -1 \ 0 & 2 & 0 \ -1 & 0 & 3 \end{array}\end{gathered}$$\right] \
\therefore \text { Now }(A-31)\left[$$\begin{gathered} x \ y \ z \end{gathered}$$\right]=\left[$$\begin{gathered} -1 \ 2 \ 3 \end{gathered}$$\right] \
\therefore\left[$$\begin{array}{ccc} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{array}$$\right]\left[$$\begin{gathered} x \ y \ z \end{gathered}$$\right]=\left[$$\begin{gathered} -1 \ 2 \ 3 \end{gathered}$$\right] \
{\left[$$\begin{gathered} -z \ -y \ -x \end{gathered}$$\right]=\left[$$\begin{gathered} 1 \ 2 \ 3 \end{gathered}$$\right]} \
{[z=-1],[y=-2],[x=-3]}
\]

Solutions · 1