Chapter Contents

• Matrices and Determinants
• 1. Determinants
• 2. Large Determinants
• 3. Matrices
• 4. Multiplication of Matrices
• Matrix Multiplication examples
• Add & multiply matrices - Flash
• 5. Finding the Inverse of a Matrix
• Multiplication and Inverse examples
• Multiplication and Inverse examples - own numbers
• Inverse of a Matrix using Gauss-Jordan Elimination
• 6. Matrices and Linear Equations
• Matrices and Determinants Problem Solver

• Comments, Questions?

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Matrix Examples - Multiplication and Inverse (2×2)

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You get a new example of matrix multiplication and inverses each time you load the page.

We are given 2 matrices, ` A = ((1,3),(4,-3)) ` and ` B = ((5,4),(8,10))`.

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` and `B` above, we have:

`A B = ((1,3),(4,-3)) ((5,4),(8,10))= ((1 xx 5 + 3 xx 8,1 xx 4 + 3 xx 10),(4 xx 5 + -3 xx 8,4 xx 4 + -3 xx 10)) = ((29,34),(-4,-14))`

`B A = ((5,4),(8,10)) ((1,3),(4,-3))= ((5 xx 1 + 4 xx 4,5 xx 3 + 4 xx -3),(8 xx 1 + 10 xx 4,8 xx 3 + 10 xx -3)) = ((21,3),(48,-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 = ((1,3),(4,-3))`

is

`A^-1 = 1/det(A)((-3,-3),(-4,1))=1/-15((-3,-3),(-4,1))=((0.2,0.2),(0.26667,-0.06667))`

Check:

`A A^-1=((1,3),(4,-3))((0.2,0.2),(0.26667,-0.06667))=((1,0),(0,1))`

`A^-1 A=((0.2,0.2),(0.26667,-0.06667))((1,3),(4,-3))=((1,0),(0,1))`

So we know we have found the correct inverse.

The inverse of `B = ((5,4),(8,10))` is

`B^-1=1/det(B)((10,-4),(-8,5))=1/18((10,-4),(-8,5))=((0.55556,-0.22222),(-0.44444,0.27778))`

Check:

`B B^-1=((5,4),(8,10))((0.55556,-0.22222),(-0.44444,0.27778))`

`=B^-1 B=((0.55556,-0.22222),(-0.44444,0.27778))((5,4),(8,10))=((1,0),(0,1))`

See another example

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