d)
Given:
[tex]Let\text{ X}=\begin{bmatrix}{2} & {4} & {1} \\ {} & {} & {} \\ {-1} & {3} & {5}\end{bmatrix}[/tex]Required:
We need to rotate the triangle by 270 degrees.
Explanation:
When rotating a point 270 degrees counterclockwise about the origin our point A(x,y) becomes A'(y, -x). This means, we switch x and y and make x negative.
Switch the first and second row of the given matrix and multiply the first row by (-1).
[tex]\text{X'}=\begin{bmatrix}{-1} & {3} & {5} \\ {} & {} & {} \\ 2 & {4} & 1\end{bmatrix}R_1\leftrightarrow R_{2\text{ }}[/tex]Multiply the first row by (-1).
[tex]\text{X'}=\begin{bmatrix}{1} & -{3} & -{5} \\ {} & {} & {} \\ 2 & {4} & 1\end{bmatrix}[/tex]Answer:
The vertices of the image of the traignle ABC is
[tex]\begin{bmatrix}{1} & -{3} & -{5} \\ {} & {} & {} \\ 2 & {4} & 1\end{bmatrix}[/tex]e)
Given:
[tex]Let\text{ X}=\begin{bmatrix}{2} & {4} & {1} \\ {} & {} & {} \\ {-1} & {3} & {5}\end{bmatrix}[/tex]Required:
We need to reflect the triangle across the y-axis.
Explanation:
When we reflect a point across the y-axis our point A(x,y) becomes A'(x,-y).
This means making a negative of x.
Multiply the first row by (-1).
[tex]\begin{bmatrix}{-2} & {-4} & {-1} \\ {} & {} & {} \\ -{1} & {3} & {5}\end{bmatrix}[/tex]Answer:
The vertices of the image of the traignle ABC is
[tex]\begin{bmatrix}{-2} & {-4} & {-1} \\ {} & {} & {} \\ -{1} & {3} & {5}\end{bmatrix}[/tex]