Mathematics- Solving Matrices the Old School Way, Sept. 19, 2008

In today’s class we recapped from the investigation yesterday. Yesterday’s investigation was meant to be an introduction to the idea of an inverse of a matrix.

Inverse of a Matrix

  • For a given matrix [A], the inverse of the matrix would be noted as [A]-1
  • Only square matrices have inverses
  • The inverse of a matrix can be found by hand or by calculator
  • When a matrix [A] is multiplied by its inverse [A]-1, the result is an identity matrix I
    [A][A]-1=I
  • Identity matrices take on the form:
    [1 0
    0 1]
    Where there is a diagonal of “1”s
  • The identity matrix is like the number 1 in matrices
  • The idenity matrix has the property AI = IA= A ; if we multiply an identity matrix with matrix A, the result will be the same as the original matrix A

Why Would We Need An Inverse?

  • If we consider the equation 3x=12, we need to isolate x for our solution
  • we divide the left side by 3 which is the equivalent of multiplying by 1/3
  • 1/3 is the inverse of 3
  • we similarly need inverses of matrices to isolate our variables; we multiply both sides by the inverse of the coefficient matrix to isolate our x, y, z…

Note: when we multiply matrices, remember that the first matrix must have as many columns as the second matrix has in rows.

Solving Matrices By Hand

1) First find what we call the determinant. For the matrix:

A = [a  b
c  d]

The determinant is det(A)= ad-bc

If det(A)= 0, there is no solution or an infinite number of solutions
If det(A) does not equal 0, there is a unique solution

2) If there is a determinant,

A-1 = (1/det(A)) [d  -b
-c   d]

“1 divided by the determinant multipled by the rearranged matrix”

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