Matrix Inverses Notes

Inverse of Matrix A

Definition of Inverse

• The inverse of a matrix A is similar to the reciprocal of a number.
• If matrix A has an inverse A^-1, it holds that:
  • A \cdot A^{-1} = I
  • Where I represents the identity matrix.
• If the determinant of the matrix |A| is zero, then the matrix does not have an inverse.

Determinant of Matrix A

• The determinant of matrix A can be calculated for a 2x2 matrix A = ( \begin{pmatrix} a & b \ c & d \end{pmatrix} ) as:

  |A| = ad - bc

Finding the Inverse of a 2x2 Matrix

• For matrix A, the inverse can be computed as:

  A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}

• For example, for the matrix ( A = \begin{pmatrix} 4 & 1 \ 6 & 2 \end{pmatrix} ):

  • Calculate its determinant:

    |A| = (4)(2) - (1)(6) = 8 - 6 = 2

  • Then the inverse is:

    A^{-1} = \frac{1}{2} \begin{pmatrix} 2 & -1 \ -6 & 4 \end{pmatrix} = \begin{pmatrix} 1 & -0.5 \ -3 & 2 \end{pmatrix}

Practice Finding Inverses

• Practice with the following matrices:
  • A = ( \begin{pmatrix} 5 & 3 \ 1 & 1 \end{pmatrix} )
  • B = ( \begin{pmatrix} 0 & -1 \ 0.5 & 2 \end{pmatrix} )
  • C = ( \begin{pmatrix} -5 & 10 \ 2 & -4 \end{pmatrix} )
  • D = ( \begin{pmatrix} -3 & 9 \ -3 & 8 \end{pmatrix} )
  • E = ( \begin{pmatrix} 7 & 4 \ 3 & 2 \end{pmatrix} )
• If the inverse does not exist for any matrix, state why not—usually due to a determinant of zero.

Decoding a Riddle Using Inverses

• Instructions:

  1. Find the inverses of the following matrices.
  2. Use the elements of the inverses to decode a riddle.

• Examples of matrices:

  • J = ( \begin{pmatrix} -3 & 4 \ 0 & 1 \end{pmatrix} )
  • K = ( \begin{pmatrix} 4 & 3 \ -1 & 1 \end{pmatrix} )
  • L = ( \begin{pmatrix} -5 & 3 \ 3 & -2 \end{pmatrix} )
  • M = ( \begin{pmatrix} -3 & -3 \ 6 & 4 \end{pmatrix} )

• Example Decoding (assuming inverses are found):
  J^-1 = R
  K^-1 = L
  L^-1 = U
  M^-1 = M