Determinants and Inverses

  1. Determinants:

    Consider row reducing the standard 2x2 matrix.  Suppose that a is nonzero.


    a b
    c d

     
    1/a R1 ->  R1 R2 - cR1  -> R2
    1 b/a
    c d
    1

    b/a
    0 d - cb/a

    Now notice that we cannot make the lower right corner a 1 if

            d - cb/a = 0 

     or

            ad - bc = 0


              Definition of the Determinant

    We call ad - bc the determinant of the 2 by 2 matrix
    a b
    c d

    it tells us when it is possible to row reduce the matrix and find a solution to the linear system.

    Example:  

    The determinant of the matrix


    3 1
    5 2

    is 

            3(2) - 1(5) = 6 - 5 = 1


  2. Determinants of Three by Three Matrices

    We define the determinant of a triangular matrix

     
    a d e
    0 b f
    0 0 c

    by 

            det = abc

    Notice that if we multiply a row by a constant k then the new determinant is k times the old one.  We list the effect of all three row operations below.

                         Theorem

    The effect of the the three basic row operations on the determinant are as follows
    1. Multiplication of a row by a constant multiplies the determinant by that constant.
    2. Switching two rows changes the sign of the determinant.
    3. Replacing one row by that row + a multiply of another row has no effect on the determinant.


    To find the determinant of a matrix we use the operations to make the matrix triangular and then work backwards.

    Example:

    Find the determinant of


    2 6 10
    2 4 -3
    0 4 2

    We use row operations until the matrix is triangular.

            1/2 R1 <-> R1     (Multiplies the determinant by 1/2)
    1 3 5
    2 4 -3
    0 4 2

      
            R2 - 2R1 -> R2     (No effect on the determinant)
    1 3 5
    0 -2 -13
    0 4 2

    Note that we do not need to zero out the upper middle number.  We only need to zero out the bottom left numbers.

    R3 + 2R2 ->  R3     (No effect on the determinant)  
    1 3 5
    0 -2 -13
    0 0 -24

    Note that we do not need to make the  middle number a 1.

    The determinant of this matrix is 48. Since this matrix has 1/2 the determinant of the original matrix, the determinant of the original matrix has 

            determinant = 48(2) = 96.

  3. Inverses

    We call the square matrix I with all 1's down the diagonal and zeros everywhere else the identity matrix.  It has the unique property that if A is a square matrix with the same dimensions then

            AI = IA = A

     

    Definition

    If A is a square matrix then the inverse A-1 of A is the unique matrix such that


    AA-1 = A-1A = I  


    Example:  

    Let 
              A =
    2 5
    1 3

    then
              A-1  =
    3 -5
    -1 2

    Verify this!

                        Theorem

    The inverse of a matrix exists if and only if the determinant is nonzero.

    To find the inverse of a matrix, we write a new extended matrix with the identity on the right.  Then we completely row reduce, the resulting matrix on the right will be the inverse matrix.

    Example:


    2 -1
    1 -1

    First note that the determinant of this matrix is 

            -2 + 1 = -1

    hence the inverse exists.  Now we set the augmented matrix as


    2 -1 1 0
    1 -1 0 1


    R1 <-> R2   R2 - 2R1 -> R2 R1 + R2  -> R1
    1 -1 0 1
    2 -1 1 0
    1 -1 0 1
    0 1 1 -2
    1 0 1 -1
    0 1 1 -2

    Notice that the left hand part is now the identity.  The right hand side is the inverse.  Hence

     
              A-1 =
    1 -1
    1 -2

     

  4. Solving Equations Using Matrices

    Example:

    Suppose we have the system

            2x - y = 3
            x - y = 4

    Then we can write this in matrix form

            Ax = b

    where
              A =
    2 -1
    1 -1
    		x =
    x
    y

    and      b =
    3
    4

    We can multiply both sides by A-1:

            A-1A x = A-1b

    or 

            x = A-1b

    From before,
              A-1 =  
    1 -1
    1 -2

    Hence our solution is

     
    -1
    -5


    or 

            x = -1     and     y = 5

     

  5. The Easy Way

    A graphing calculator can be used to work all of the above problems. 

 

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