How do I solve this matrix equation?

Question

How do I solve this matrix equation and what is the answer:

$$\begin{bmatrix} -122.366667 \\ 37.61666667 \end{bmatrix} = \begin{bmatrix} 0.000046 & 0.000032 & -122.413307 \\ 0.000025 & -0.000036 & 37.632195 \end{bmatrix}\begin{bmatrix} x \\ y \\ 1 \end{bmatrix}$$

This doesn't seem to be a matrix-vector equation of the conventional sort... — J. M. ain't a mathematician, Oct 26 '11 at 23:18
You can try striking out the last column of the coefficient matrix and last row of the variable vector, yielding a usual 2-by-2 system. See if the solution of that system satisfies your original system... — J. M. ain't a mathematician, Oct 26 '11 at 23:30

score 2 · Answer 1 · edited Feb 24 '12 at 09:28

2

In general, to solve for $C$ when you have $A = BC$ and they're all matrices, you have to:

find the inverse of $B$, call it $B^{-1}$
left-multiply each side by $B^{-1}$ to get $B^{-1}A = C$

edited Feb 24 '12 at 09:28

Asaf Karagila

393,674

answered Oct 26 '11 at 23:29

Kevin Lacker

21

And if $B$ isn't invertible? – Oct 27 '11 at 00:38

smackcrane · Accepted Answer · 2011-10-26T23:58:04.737

1

We can write out the matrix multiplication to get a system of equations, and then solve that. Here is the expanded matrix multiplication: $$ 0.000046x + 0.000032y - 122.413307 = -122.366667 $$$$ 0.000025x - 0.000036y + 37.632195 = 37.61666667 $$ So we end up with two equations of two variables, which can be easily solved by a method of your choice (substitution, elimination, matricies, etc). $$ x\approx 481.325, y\approx 765.596 $$ as you should verify.

edited Oct 26 '11 at 23:58

answered Oct 26 '11 at 23:32

smackcrane

3,211

You seems to have lost the decimal point in -122.366667. This may make a difference. – Henry Oct 26 '11 at 23:50
@Henry many thanks, corrected. This is why it's important to verify answers :) – smackcrane Oct 27 '11 at 00:00

score 1 · Answer 3 · answered Oct 26 '11 at 23:41

Note that, this can be brought down to the following square system since 1 is not a variable hence can be included at the left hand side. (Just multiply everything to see this.)

$$\pmatrix{ 122.413307-122.366667 \\ -37.632195 + 37.61666667 } = \pmatrix{0.046640\\ -0.01552833}=\begin{bmatrix} 0.000046 & 0.000032\\ 0.000025 & -0.000036 \end{bmatrix}\pmatrix{ x \\ y} $$ Now you are back at the $b=Az$ form which can be solved by multiplying both sides by the matrix $A^{-1}$ to obtain $z=A^{-1}b$. Again, this only makes sense if your matrix $A$ is invertible if not you have to apply further steps.

$$ z = A^{-1}b = \pmatrix{ 14657.9804560261 & 13029.3159609121\\ 10179.1530944625 &-18729.6416938111 }\pmatrix{0.046640\\ -0.01552833} \approx \pmatrix{481.3 \\765.6} $$

I did not pay attention to the significant digits (and it felt good!) hence you have to take care of that.

How do I solve this matrix equation?

3 Answers3