# Matrix Examples - Multiplication and Inverse (2×2)

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We are given 2 matrices, ` A = ((6,8),(9,2)) ` and ` B = ((2,1),(5,7))`.

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 = ((6,8),(9,2)) ` and ` B = ((2,1),(5,7))` given above, we have:

`{: (A B, = ((6,8),(9,2)) ((2,1),(5,7)) ), (, = ((6 xx 2 + 8 xx 5,6 xx 1 + 8 xx 7),(9 xx 2 + 2 xx 5,9 xx 1 + 2 xx 7)) ), (, = ((52,62),(28,23)) ) :} `

`{: (B A, = ((2,1),(5,7)) ((6,8),(9,2)) ), (,= ((2 xx 6 + 1 xx 9,2 xx 8 + 1 xx 2),(5 xx 6 + 7 xx 9,5 xx 8 + 7 xx 2)) ), (, = ((21,18),(93,54)) ) :}`

## 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 = ((6,8),(9,2))`

is

`A^-1 = 1/det(A)((2,-8),(-9,6))=1/-60((2,-8),(-9,6))=((-0.03333,0.13333),(0.15,-0.1))`

Check:

`A A^-1=((6,8),(9,2))((-0.03333,0.13333),(0.15,-0.1))=((1,0),(0,1))`

`A^-1 A=((-0.03333,0.13333),(0.15,-0.1))((6,8),(9,2))=((1,0),(0,1))`

So we know we have found the correct inverse.

The inverse of `B = ((2,1),(5,7))` is

`B^-1=1/det(B)((7,-1),(-5,2))=1/9((7,-1),(-5,2))=((0.77778,-0.11111),(-0.55556,0.22222))`

Check:

`B B^-1=((2,1),(5,7))((0.77778,-0.11111),(-0.55556,0.22222))=((1,0),(0,1))`

`B^-1 B=((0.77778,-0.11111),(-0.55556,0.22222))((2,1),(5,7))=((1,0),(0,1))`

So

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

Our inverse is correct.

See Easy math input and nice output using ASCIIMathML and MathJax for how the math on this page is being displayed.

ASCIIMathTexImg.js © David Lippman, Pierce College at Ft Steilacoom.

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