Matrix Examples - Multiplication and Inverse (2×2)
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We are given 2 matrices, ` A = ((-4,-2),(-1,-8)) ` and ` B = ((0,-1),(3,5))`.
We'll use these matrices for the following examples.
2×2 Matrix Multiplication
In general, if `X = ((a,b),(c,d))` and `Y = ((e,f),(g,h))`, then the multiplication `X Y` is given by:
`X Y = ((a,b),(c,d)) ((e,f),(g,h)) = ((a xx e + b xx g,a xx f + b xx h),(c xx e + d xx g,c xx f + d xx h)) `
So for our matrices `A` and `B` above, we have:
`A B = ((-4,-2),(-1,-8)) ((0,-1),(3,5))= ((-4 xx 0 + -2 xx 3,-4 xx -1 + -2 xx 5),(-1 xx 0 + -8 xx 3,-1 xx -1 + -8 xx 5)) = ((-6,-6),(-24,-39)) `
` B A = ((0,-1),(3,5)) ((-4,-2),(-1,-8))= ((0 xx -4 + -1 xx -1,0 xx -2 + -1 xx -8),(3 xx -4 + 5 xx -1,3 xx -2 + 5 xx -8)) = ((1,8),(-17,-46)) `
2×2 Matrix Inverse
In general, the inverse of the 2×2 matrix `X = ((a,b),(c,d))` is given by:
` X^-1 = 1/det(X) ((d, -b),(-c,a))`
Note: This only works for 2×2 matrices.
So for matrices `A` and `B` given above, we have the following results.
The inverse of
`A = ((-4,-2),(-1,-8))`
is
`A^-1 = 1/det(A) ((-8,2),(1,-4)) = 1/30((-8,2),(1,-4)) = ((-0.26667,0.06667),(0.03333,-0.13333))`
Check:
` A A^-1 = ((-4,-2),(-1,-8)) ((-0.26667,0.06667),(0.03333,-0.13333)) `
` = A^-1 A = `
` ((-0.26667,0.06667),(0.03333,-0.13333)) ((-4,-2),(-1,-8)) = ((1, 0), (0, 1)) `
The inverse of `B = ((0,-1),(3,5))` is
`B^-1 = 1/det(B) ((5,1),(-3,0)) = 1/3((5,1),(-3,0)) = ((1.66667,0.33333),(-1,0))`
Check:
`B B^-1 = ((0,-1),(3,5)) ((1.66667,0.33333),(-1,0))`
` = B^-1 B = ((1.66667,0.33333),(-1,0)) ((0,-1),(3,5)) = ((1, 0), (0, 1))`
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