# Simplfy a complex matrix into a real one

I encounter systems of linear complex equations (At most 3 equations) in my circuit analysis course. The calculator I am using is Casio fx-991ES and it only accepts real elements when in matrix or equation solver mode. So, is there a way to manipulate a matrix so that there are no $j$'s left in it?

The only solution I found was replacing each element $a+bj$ with a 2 by 2 matrix. \begin{array}{ccc} a & -b\\ b & a \\ \end{array}

but then the matrix dimensions will be doubled making a 2x2 matrix a 4x4 matrix. which unfortunately exceeds the casio 3x3 limit for maximum matrix size.

The only other solution is using Cramer's rule which works well for 2x2 matrices but gets really messy for 3x3.

So, is there a solution from the theoretical side other than those two ?

If not would you please suggest a calculator that is capable of solving complex equations? my college's policy is so strict regarding this matter all graphing and sophisticated programmable calculators are not allowed. If there is a calculator close to 991ES that would be great.

Thanks in advanced.

## 2 Answers

You could try storing two separate matrices for the real and imaginary components. So for example, $$\pmatrix{ 1 + j & 2-j\\ 4 & 3j } = \pmatrix{ 1&2\\4&0 } +j\pmatrix{ 1 & -1\\0 &3 }$$ Suppose that $M = A + jB$ and $N = C + jD$. It would follow that $$MN = (A + jB)(C + jD) = (AC - BD) + j(AD + BC)$$

• And then where to go from that? If this matrix A was multiplied by a 2x1 matrix X like AX=B and be is given how can I find X? after splitting A and B into two matrices ? – Loers Antario Mar 15 '14 at 11:13
• @LoersAntario see my latest edit. This will work as long as $M$ and $N$ have compatible dimensions, i.e. as long as they can be multiplied. – Ben Grossmann Mar 15 '14 at 14:07

I don't think that there can be any other way for that! So you need to find another calculator, but I would prefer a program rather than a calculator. Matlab is a very good program for working with matrices, and it doesn't have such a difficult syntax.

• We are not allowed to use Matlab in exams. – Loers Antario Mar 14 '14 at 23:17
• I didn't knew that you are talking about being in exam! Well, than you need to ask this question to your professor! He might help you with that. – Emo Mar 14 '14 at 23:19
• Sorry, I should've clarified that in the question, and my professors aren't so cooperative. If they were they could've just allowed us to use any better calculator. – Loers Antario Mar 14 '14 at 23:32