**Unformatted text preview: **omials involving matrices. The reader is
encouraged to step back and compare our expansion of the matrix product (M − 2I4 ) (M + 3I4 ) in
part 3 with the product (x − 2)(x + 3) from real number algebra. The exercises explore this kind
of parallel further.
As we mentioned earlier, a point P (x, y ) in the xy -plane can be represented as a 2 × 1 position
matrix. We now show that matrix multiplication can be used to rotate these points, and hence
graphs of equations.
√ Example 8.3.3. Let R = 2
2
√
2
2 √ − 2
2
√
2
2 . 1. Plot P (2, −2), Q(4, 0), S (0, 3), and T (−3, −3) in the plane as well as the points RP , RQ,
RS , and RT . Plot the lines y = x and y = −x as guides. What does R appear to be doing
to these points?
2
2. If a point P is on the hyperbola x2 − y 2 = 4, show that the point RP is on the curve y = x . 486 Systems of Equations and Matrices Solution. For P (2, −2), the position matrix is P = 2
, and
−2 √ RP =
= 2
2
√
2
2
√ √ − 2
2
√
2
2 2
−2 22
0 √
√√
We have that R √ √(2, −2) to (2 2, 0). Similarly, we ﬁnd (4, 0) is moved to (2 2, 2 2), (0, 3)
takes
...

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