Question

How do you solve `((2, 0, 0), (-1, 2, 0), (-2, 4, 1))x=((4), (10), (11))`?

Answer 1

`x = ((2), (6), (-9))`

Explanation:

First construct the inverse matrix of `((2, 0, 0), (-1, 2, 0), (-2, 4, 1))` by writing the identity matrix alongside it and transforming that as we transform the original matrix into the identity:

`((2, 0, 0, |, 1, 0, 0), (-1, 2, 0, |, 0, 1, 0), (-2, 4, 1, |, 0, 0, 1))`

Add row `1` to row `3`:

`((2, 0, 0, |, 1, 0, 0), (-1, 2, 0, |, 0, 1, 0), (0, 4, 1, |, 1, 0, 1))`

Divide row `1` by `2`:

`((1, 0, 0, |, 1/2, 0, 0), (-1, 2, 0, |, 0, 1, 0), (0, 4, 1, |, 1, 0, 1))`

Add row `1` to row `2`:

`((1, 0, 0, |, 1/2, 0, 0), (0, 2, 0, |, 1/2, 1, 0), (0, 4, 1, |, 1, 0, 1))`

Subtract `2 xx` row `2` from row `3`:

`((1, 0, 0, |, 1/2, 0, 0), (0, 2, 0, |, 1/2, 1, 0), (0, 0, 1, |, 0, -2, 1))`

Divide row `2` by `2`:

`((1, 0, 0, |, 1/2, 0, 0), (0, 1, 0, |, 1/4, 1/2, 0), (0, 0, 1, |, 0, -2, 1))`

So:

`((2, 0, 0), (-1, 2, 0), (-2, 4, 1))^(-1) = ((1/2, 0, 0), (1/4, 1/2, 0), (0, -2, 1))`

Then multiply the right hand side column matrix by our inverse matrix to find:

`x = ((1/2, 0, 0), (1/4, 1/2, 0), (0, -2, 1))((4),(10),(11)) = ((2), (6), (-9))`

Answer 2

`x = ((2), (6), (-9))`

Explanation:

Alternatively, don't bother to construct an inverse matrix: Just perform a similar sequence of steps with the target column matrix appended to our original matrix as follows:

`((2, 0, 0, |, 4 ), (-1, 2, 0, |, 10), (-2, 4, 1, |, 11))`

Add row `1` to row `3`:

`((2, 0, 0, |, 4 ), (-1, 2, 0, |, 10), (0, 4, 1, |, 15))`

Divide row `1` by `2`:

`((1, 0, 0, |, 2 ), (-1, 2, 0, |, 10), (0, 4, 1, |, 15))`

Add row `1` to row `2`:

`((1, 0, 0, |, 2 ), (0, 2, 0, |, 12), (0, 4, 1, |, 15))`

Subtract `2 xx` row `2` from row `3`:

`((1, 0, 0, |, 2 ), (0, 2, 0, |, 12), (0, 0, 1, |, -9))`

Divide row `2` by `2`:

`((1, 0, 0, |, 2 ), (0, 1, 0, |, 6), (0, 0, 1, |, -9))`

Having reached the identity matrix on the left hand side, we can read off the solution from the right hand side:

`x = ((2), (6), (-9))`