The 2 x 2 matrices of the transformations are
[tex]\left[\begin{array}{cc}-1&0\\0&1\end{array}\right][/tex], [tex]\left[\begin{array}{cc}0&1\\1&0\end{array}\right][/tex], [tex]\left[\begin{array}{cc}-1&0\\0&-1\end{array}\right][/tex] and [tex]\left[\begin{array}{cc}\lambda&0\\0&\lambda\end{array}\right][/tex]
The rule of reflection in the y-axis is
(x, y) ⇒ (-x, y)
This is represented as:
[tex]\left[\begin{array}{cc}a&b\\c&d\end{array}\right] \left[\begin{array}{c}x&y\end{array}\right] = \left[\begin{array}{c}-x&y\end{array}\right][/tex]
When solved using a graphing calculator, we have:
[tex]\left[\begin{array}{cc}-1&0\\0&1\end{array}\right] \left[\begin{array}{c}x&y\end{array}\right] = \left[\begin{array}{c}-x&y\end{array}\right][/tex]
Hence, the 2 x 2 matrix is [tex]\left[\begin{array}{cc}-1&0\\0&1\end{array}\right][/tex]
The rule of reflection in the line y = x is
(x, y) ⇒ (y, x)
This is represented as:
[tex]\left[\begin{array}{cc}a&b\\c&d\end{array}\right] \left[\begin{array}{c}x&y\end{array}\right] = \left[\begin{array}{c}y&x\end{array}\right][/tex]
When solved using a graphing calculator, we have:
[tex]\left[\begin{array}{cc}0&1\\1&0\end{array}\right] \left[\begin{array}{c}x&y\end{array}\right] = \left[\begin{array}{c}y&x\end{array}\right][/tex]
Hence, the 2 x 2 matrix is [tex]\left[\begin{array}{cc}0&1\\1&0\end{array}\right][/tex]
The rule of rotation through 180◦ about the origin is
(x, y) ⇒ (-x, -y)
This is represented as:
[tex]\left[\begin{array}{cc}a&b\\c&d\end{array}\right] \left[\begin{array}{c}x&y\end{array}\right] = \left[\begin{array}{c}-x&-y\end{array}\right][/tex]
When solved using a graphing calculator, we have:
[tex]\left[\begin{array}{cc}-1&0\\0&-1\end{array}\right] \left[\begin{array}{c}x&y\end{array}\right] = \left[\begin{array}{c}-x&y\end{array}\right][/tex]
Hence, the 2 x 2 matrix is [tex]\left[\begin{array}{cc}-1&0\\0&-1\end{array}\right][/tex]
The rule of the enlargement from the origin with scale factor λ.
(x, y) ⇒ (λx, λy)
This is represented as:
[tex]\left[\begin{array}{cc}a&b\\c&d\end{array}\right] \left[\begin{array}{c}x&y\end{array}\right] = \left[\begin{array}{c}\lambda x&\lambda y\end{array}\right][/tex]
When solved using a graphing calculator, we have:
[tex]\left[\begin{array}{cc}\lambda&0\\0&\lambda\end{array}\right] \left[\begin{array}{c}x&y\end{array}\right] = \left[\begin{array}{c}\lambda x&\lambda y\end{array}\right][/tex]
Hence, the 2 x 2 matrix is [tex]\left[\begin{array}{cc}\lambda&0\\0&\lambda\end{array}\right][/tex]
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