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)

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

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

So for our matrices ` A = ((-4,-2),(-1,-8)) ` and ` B = ((0,-1),(3,5))` given 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))=((1,0),(0,1))`

`A^-1 A=((-0.26667,0.06667),(0.03333,-0.13333))((-4,-2),(-1,-8))=((1,0),(0,1))`

So we know we have found the correct inverse.

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))=((1,0),(0,1))`

`B^-1 B=((1.66667,0.33333),(-1,0))((0,-1),(3,5))=((1,0),(0,1))`

So

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

Our inverse is correct.

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