Question

Given `A=((-1, 2), (3, 4))` and `B=((-4, 3), (5, -2))`, how do you find A-2B?

Answer 1

Follow the order of operations to find:
`A-2B=((-1, 2), (3, 4))-((-8, 6), (10, -4)) = ((7, -4), (-7, 8))`

Explanation:

To solve a matrix equation we follow the normal order of operations with the added restriction that multiplication and division need to happen in the order that they are written, since for matrices, `AB !=BA` in general (there are special cases where this is true).

So for our equation, `A-2B`, we need to start with the multiplication `2B`. Multiplying a scalar, `2`, by a matrix, `B`, has the effect of multiplying each of the matrix elements by the scalar, therefore,

`2B = 2*((-4, 3), (5, -2)) = ((-8, 6), (10, -4))`

subtracting two matrices requires that each matrix have the same dimensions - this is true in our case since `A` and `B` are both `2xx2` matrices. Subtracting matrices results in the subtraction of each element from the corresponding element in the other matrix, i.e.

`((a_(11), a_(12)), (a_(21), a_(22)))-((b_(11), b_(12)), (b_(21), b_(22))) = ((a_(11)-b_(11), a_(12)-b_(12)), (a_(21)-b_(21), a_(22)-b_(22)))`

in our case

`A-2B=((-1, 2), (3, 4))-((-8, 6), (10, -4)) = ((7, -4), (-7, 8))`