Solve for X (Linear Algebra)

Question

I keep getting this question wrong and I have no idea how to do this. It would be really appreciated if you can help me. Here's the link to the example: https://ibb.co/pjTMyg2

Answer 1

Matrices have unique properties that need to be considered.

In the example you have let:

-- --

| 2 -3 |

A = | |

| -6 -6 |

-- --

-- --

| -4 -8 |

B = | |

| 1 9 |

-- --

-- --

| 6 -9 |

C = | |

| -1 4 |

-- --

The problem then is:

[ A ] • [ X ] + [ B ] = [ C ] • [ X ]

The commutative property of addition is valid for matrices therefore we can subtract [ B ] from both sides to get:

[ A ] • [ X ] = [ C ] • [ X ] - [ B ]

To find - [ B ] you must multiply each element by -1

Again, since the commutative property of addition is valid for matrices we can subtract [ C ] • [ X ] from both sides to get:

[ A ] • [ X ] - [ C ] • [ X ] = - [ B ]

And since the distribution property works as follows:

[ A ] • [ X ] - [ C ] • [ X ] = ( [ A ] - [ C ] ) • [ X ]

we can rewrite our equation as:

( [ A ] - [ C ] ) • [ X ] = - [ B ]

You can now multiply both sides by the inverse of ( [ A ] - [ C ] ) to get the answer:

( [ A ] - [ C ] )^-1 • ( [ A ] - [ C ] ) • [ X ] = ( [ A ] - [ C ] )^-1 • (- [ B ])

of course a matrix multiplied by its inverse wipes out both so we get

[ X ] = ( [ A ] - [ C ] )^-1 • (- [ B ])

You can plug this into a calculator that does matrix operations or do it by hand but this provides the answer.

The order things are in makes a difference. Notice the distribution property maintains the order of the matrices (A comes first followed by C and then multiplied by X just like it was prior to distributing. Also, when we multiply the inverse on both sides it is first on the left side and first on the right side.