hi there would anyone be able to give me a link or guidence as how i would go about finding the inverse of A? I have a book that tells me add subtract and multiply. the question is this: find the inverse of A A = (19 81) (2 10) sorry for the crude matrices. any help would be great :) lakitu
there is a trick for finding the inverse of a 2x2 matrix [tex]A=\left( \begin{array}{cc} a & b \\ c & d \end{array} \right)[/tex] then [tex]A^{-1}=\frac{1}{ad-bc}\left( \begin{array}{cc} d & -b \\ -c & a \end{array} \right)[/tex] provided ad-bc is not equal to zero hope that helps
http://www.purplemath.com/modules/mtrxinvr.htm halfway down is how i usually find matrices. always works, so thats what i use.
If your book is telling you "add, subtract, and multiply" (hey, that's how you solve any mathematics problem! ) then go back and read over exactly what you add and subtract and what you multiply by. I suspect that your book is talking about "row operations"- that's what Gale's website is talking about.
Before rushing off to a formula or procedure, it might be helpful to understand what you are trying to do when determining the inverse of a matrix. Given a square matrix [tex]A=\left( \begin{array}{cc} a & b \\ c & d \end{array} \right)[/tex], its inverse [if it exists] is another square matrix [tex]A^{-1}=\left( \begin{array}{cc} p & q \\ r & s \end{array} \right)[/tex] such that the matrix product is the identity matrix. [tex] \begin{align*} AA^{-1}&=I\\ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \left( \begin{array}{cc} p & q \\ r & s \end{array} \right) &= \left( \begin{array}{cc} 1 & 0 \\ 0 & 1\end{array} \right) \end{align*} [/tex] If you carry out the matrix multiplication, you should find a simple system of four linear equations in four unknowns... "simple" because it's really a pair of systems of two linear equations in two unknowns. You can easily solve these systems to obtain the formula given by vladimir69 above. (In addition, the inverse would satisfy [tex]A^{-1}A&=I[/tex] as well.)
Of course, if you have a 3 by 3 or 5 by 5 matrix, so that your system is 9 equation in 9 unknowns or 25 equations in 25 unknowns, you might find robphy's method a bit tedious! I think it's worth learning row reduction.
This is how I like to think of it. If this doesn't make sense to you, feel free to forget about it so you don't get confused! We start off with the partitioned matrix: [A : I] which has the property that: (the left matrix) = (the right matrix) * A. In particular, A = I*A. Now, if we do row operations, we will get some other partitioned matrix: [B : C] which still has the property that: (the left matrix) = (the right matrix) * A. In particular, B = C*A If we fully row-reduce the left hand side, we get the partitioned matrix: [I : V] which still has the property that (the left matrix) = (the right matrix) * A. In particular, I = V*A, and therefore V is the inverse of A.