Inverse matrix

What is the inverse matrix?

Every matrix that isn’t singular has an inverse matrix. Whenever we multiply a matrix by its inverse matrix, we get the identity matrix, .

Representing the inverse matrix

• The inverse matrix of can be written as .
• This then makes sense why they multiply to the identity matrix:
  • anything multiplied by its inverse is equal to the identity, or if we’re working with regular scalars.

Finding the inverse matrix

Find the inverse matrix of

• Solve the equations with and simultaneously:
  • (1)
  • (2)
  • Multiply (1) by : (3)
  • Multiply (2) by : (4)
  • Solve using elimination:
    • (3) + (4):
  • Substitute into (1):
  • ,

General form of the inverse matrix

• Let
• Find the determinant:

Find the inverse of matrix product

• Let the inverse function be
• (because matrix multiplication is associative)
• (pre-multiply both sides by $A^{-1})
• **Answer:

Remember this. THe inverse of .