Matrix inverses

• The inverse of a matrix is , and plays the same role as does for in matrix multiplication
  • We can use the inverse of a matrix to solve systems of linear equations
  • The only number that does not have an inverse is 0, but many matrices do not have a inverse

Definition: Left and right inverses of matrices. Let be a matrix. If there is another matrix such that

BA = I $$> then is called a left inverse of . If there is another matrix such that

AC = I $$> then is called a right inverse of

• For example the matrix has no inverse, as there are no products that multiply and add to 1. to get the identity matrix.
  • For , for , If doesn’t have linearly independent columns, it doesn’t have a matrix

Proposition: If a matrix is a rectangular matrix, it has at most 1 inverse, with the number of columns in the inverse being the number of linearly independent columns in the matrix Example: Let . We knew that this has at most 2 linearly independent column vectors, so it only has 1 inverse with 2 columns. We can calculate the inverse to be . This is the right inverse of . We know it must not have a left inverse. Say that it did. We know that it must satisfy the equation . Solving this system of linear equations, we get that , and , which cannot be true.

• Remark: If is a full rank, the one of the inverses exist.
• A matrix that has both a left and right inverse is called an invertible matrix.

Definition: Invertible matrix. An invertible matrix is a matrix that has both a left inverse and a right inverse.

• If a matrix has both a left and right inverse, they are identical, and such a matrix is called the inverse of and is denoted .
  • We can arrive at this result using the associative property
    • and , so
• There is a formula for the inverse of 2 x 2 matrices: the inverse of

• A 2 x 2 matrix is invertible is
• If and are both invertible, then is invertible, and the inverse is