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.