This is false (kinda). It is true you cannot divide. However, it probably doesn't mean what you think it means.

Imagine I gave you this equation:

5x = 10

And I want you to solve for x. No problem, right? Well, what if I told you you cannot divide. What now?

Well, that is simple too. You can just multiply by the reciprocal. So instead of doing 10/5, you can do 10*(0.2). It is a trivial change. Your answer is still x = 2.

So what if I said:

Ax = B

Solve for x without dividing. Now you need to know what the reciprocal of A is. In general, the reciprocal of any number is A^(-1). So we get that:

x = B * A^(-1)

Great! Now we can say that:

[A]x = [B]

x = [A]^(-1) * [B] <-- must be in this order

But this isn't really the reciprocal. Because what I said earlier was kind of cheating. Because 1/B is still division, perse; I just wanted the notation of ^(-1). Sure, when applied to integers and such, it does give a reciprocal. However, when applied to matrices, it gives the inverse matrix.

But what is this inverse matrix nonsense? You know how if you multiply any number by its reciprocal you get 1? Well this inverse matrix is about the same; if you multiply a matrix by its inverse matrix, you get the identity matrix (where it's filled with 0's and 1's and where the 1's form a diagonal line). Similar, eh?

Inverse-matrix-finding is easy by hand if it is a 2x2. If it is 3x3 or larger (as it must be a square), it can be fairly difficult; your calculator can do it quickly though. But for a 2x2 matrix:

[a b]

[c d]

rewrite it as:

[d -b]

[-c a]

Then the determinant R = (d)(a) - (-c)(-b)

And the inverse matrix is:

[d/R -b/R]

[-c/R a/R]

Ya, I know, it's a lot of work. Because even after that, you still need to multiply the thing to another matrix.

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