Matrix Examples - Multiplication and Inverse (2×2)

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We are given 2 matrices, ` A = ((0,2),(3,-4)) ` and ` B = ((-4,-5),(-1,1))`.

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)) = ((ae + bg,af + bh),(ce + dg,cf + dh)) `

So for our matrices ` A = ((0,2),(3,-4)) ` and ` B = ((-4,-5),(-1,1))` given above, we have:

`{: (A B, = ((0,2),(3,-4)) ((-4,-5),(-1,1)) ), (, = ((0 xx -4 + 2 xx -1,0 xx -5 + 2 xx 1),(3 xx -4 + -4 xx -1,3 xx -5 + -4 xx 1)) ), (, = ((-2,2),(-8,-19)) ) :} `

`{: (B A, = ((-4,-5),(-1,1)) ((0,2),(3,-4)) ), (,= ((-4 xx 0 + -5 xx 3,-4 xx 2 + -5 xx -4),(-1 xx 0 + 1 xx 3,-1 xx 2 + 1 xx -4)) ), (, = ((-15,12),(3,-6)) ) :}`

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 = ((0,2),(3,-4))`

is

`A^-1 = 1/det(A)((-4,-2),(-3,0))=1/-6((-4,-2),(-3,0))=((0.66667,0.33333),(0.5,0))`

Check:

`A A^-1=((0,2),(3,-4))((0.66667,0.33333),(0.5,0))=((1,0),(0,1))`

`A^-1 A=((0.66667,0.33333),(0.5,0))((0,2),(3,-4))=((1,0),(0,1))`

So we know we have found the correct inverse.

The inverse of `B = ((-4,-5),(-1,1))` is

`B^-1=1/det(B)((1,5),(1,-4))=1/-9((1,5),(1,-4))=((-0.11111,-0.55556),(-0.11111,0.44444))`

Check:

`B B^-1=((-4,-5),(-1,1))((-0.11111,-0.55556),(-0.11111,0.44444))=((1,0),(0,1))`

`B^-1 B=((-0.11111,-0.55556),(-0.11111,0.44444))((-4,-5),(-1,1))=((1,0),(0,1))`

So

`BB^-1 = B^-1B = I`

Our inverse is correct.


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