Sec1Sess11

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Inverse matrix:
Inverse of A: Matrix M
Property that AM = I and MA = I

Need A to be a square matrix
<math>M\ =\ A^{-1}</math>

Linear System:
A . X = B

Linear system related reference: http://www.math.umd.edu/~petersd/460/linsyst460.pdf

Solution to A . X = B
is <math>X = A^{-1}\ .\ B</math>

Solving system of equations:
AX = B
<math>A^{-1}\ .\ (A\ .\ X) = A^{-1}\ .\ B</math>
therefore <math>X = A^{-1}\ .\ B</math> when <math>A^{-1}\ </math> cancels A

Inverse matrix <math>= A^{-1} = 1 / det(A)</math>
"adjoint matrix" = adj(A)

3x3 Matrix example steps
<math>A = \langle 2 \ 3 \ 3 \ ; \ 2 \ 4 \ 5 \ ; \ 1 \ 1 \ 2 \ \rangle </math>

Find inverse of 3x3 matrix:

Step 1
MATRIX MINORS
(same as how determinants calculate them)
<4 5; 1 2> = 4*2-5*1=3
<2 5; 1 2> = -1
<3 3; 1 2> = 3
...
end result: <3 -1 -2; 3 1 -1; 3 4 2>

Step 2
COFACTORS
Flip signs in checkerboard pattern
+ - +
- + -
+ - +

Flip the sign: "-" means flip the sign of spot in checkerboard

Step 3
TRANSPOSE
Switch rows and columns
(swap rows and columns)

Step 4
DIVIDE BY DETERMINANT OF A

step 4 only works if <math> det(A) </math> <> 0, therefore <math> det(A) </math> <> 0 is a required condition

In General:
A . x = b
M(Ax) = Mb
x = Mb

This can be used to solve linear equations such as
<math>x_1 +(2*x_2) = -1</math>
<math>(2*x_1)+(3*x_2) = 4</math>

Using:
M(Ax) = (MA)x
= Ix
= x

For examples see http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-b-matrices-and-systems-of-equations/session-11-matrix-inverses/MIT18_02SC_MNotes_m2.pdf

Finding the inverse of a 2 × 2 matrix:
(i) Switch a and d. (ii) Change the signs on b and c. (iii) Divide by the determinant.