# Find a matrix that does this

1. Feb 3, 2008

### sara_87

1. The problem statement, all variables and given/known data

find a single matrix that performs the indicated successions of operations:
expands by a factor of 5 in the y-direction, then shears with factor 2
in the y-direction

2. Relevant equations

3. The attempt at a solution

first for the expansion:
(x,y) maps to (x,5y)
then for the shear:
(x,5y) maps to (x, (2x+5y)

i think its right but im not quite sure.
thank you.

2. Feb 3, 2008

### HallsofIvy

Staff Emeritus
You have written down the transformations correctly but you haven't answered the question! You were asked to find a matrix. What matrix does that?

What matrix changes (x,y) to (x, 2x+ 5y)? In other words, find the a, b, c,d such that
$$\left[\begin{array}{cc} a & b \\ c & d\end{array}\right]\left[\begin{array}{c} x \\ y\end{array}\right]= \left[\begin{array}{c} x \\ 2x+ 5y\end{array}\right]$$
Multiplying the left side will give you two equations for a, b, c, d but remember they must be true for all x and y. Comparing corresponding coefficients will give you four very simple equations for the a, b, c, d.

Last edited: Feb 3, 2008
3. Feb 3, 2008

### sara_87

ok so multipying the left side gives the matrix:
[(ax+by),(cx+dy)]
so ax+by=x
cx+dy=2x+5y

now what? ;)

4. Feb 3, 2008

### Big-T

You might separate the equations, in order to get:
ax=x, by=0, cx=2x and dy=5y, which ought to be solvable.