Question

If you multiply the

[ 2 6 ]
[ 4 12]
​ on the left by any vector
[x]
[y]
​on the right, the resulting vector will be on what line in the plane? Give the equation of the line. [Hint: Try this out with a few examples to see where the resulting vectors are.] Answer: y=

Answer 1

The equation of the line y = 2x.

When you multiply the matrix [ 2 6 ] [ 4 12 ] on the left by any vector [x] [y] on the right, the resulting vector will be on a specific line in the plane.

To find the equation of this line, let's multiply the matrix by a few examples of vectors and observe the resulting vectors:

Example 1:

[ 2 6 ] [x] [2x + 6y]

[ 4 12] [y] = [4x + 12y]

Example 2:

[ 2 6 ] [a] [2a + 6b]

[ 4 12] [b] = [4a + 12b]

By multiplying the matrix by different vectors, we can see that the resulting vector has the form [2k + 6m] [4k + 12m], where k and m are constants. This means that the resulting vector lies on a line with the equation y = 2x.

So, the equation of the line on which the resulting vectors lie is y = 2x.

It's important to note that when you multiply the matrix [ 2 6 ] [ 4 12 ] on the left by any vector [x] [y] on the right, the resulting vector will always be on the line y = 2x.