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

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We are given 2 matrices,  A = ((-2,2),(1,-6))  and  B = ((-6,-7),(-3,-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 = ((-2,2),(1,-6))  and  B = ((-6,-7),(-3,-1)) given above, we have:

{: (A B, = ((-2,2),(1,-6)) ((-6,-7),(-3,-1)) ), (, = ((-2 xx -6 + 2 xx -3,-2 xx -7 + 2 xx -1),(1 xx -6 + -6 xx -3,1 xx -7 + -6 xx -1)) ), (, = ((6,12),(12,-1)) ) :}

{: (B A, = ((-6,-7),(-3,-1)) ((-2,2),(1,-6)) ), (,= ((-6 xx -2 + -7 xx 1,-6 xx 2 + -7 xx -6),(-3 xx -2 + -1 xx 1,-3 xx 2 + -1 xx -6)) ), (, = ((5,30),(5,0)) ) :}

## 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 = ((-2,2),(1,-6))

is

A^-1 = 1/det(A)((-6,-2),(-1,-2))=1/10((-6,-2),(-1,-2))=((-0.6,-0.2),(-0.1,-0.2))

Check:

A A^-1=((-2,2),(1,-6))((-0.6,-0.2),(-0.1,-0.2))=((1,0),(0,1))

A^-1 A=((-0.6,-0.2),(-0.1,-0.2))((-2,2),(1,-6))=((1,0),(0,1))

So we know we have found the correct inverse.

The inverse of B = ((-6,-7),(-3,-1)) is

B^-1=1/det(B)((-1,7),(3,-6))=1/-15((-1,7),(3,-6))=((0.06667,-0.46667),(-0.2,0.4))

Check:

B B^-1=((-6,-7),(-3,-1))((0.06667,-0.46667),(-0.2,0.4))=((1,0),(0,1))

B^-1 B=((0.06667,-0.46667),(-0.2,0.4))((-6,-7),(-3,-1))=((1,0),(0,1))

So

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

Our inverse is correct.

See another example

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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