Section 8
Section 8.3: Guided Notes
The Inverse of a Square Matrix
I. The Inverse of a Matrix
• We want to solve systems of equations by expressing them as
matrix equations, then solve as if we were solving ax = b; i.e. multiplying both sides by the inverse of a.
• The inverse of an n [pic] n matrix A is, if it exists, the n [pic] n matrix
A[pic] such that AA[pic]= A[pic]A = I[pic].
Example 1. Show that B is the inverse of A, where
[pic]
II. Finding Inverse Matrices
• The process that we will state shortly comes from the following:
Example 2. Find the inverse of
[pic]
We need to find the matrix
[pic]
such that AB = I[pic]. By multiplying we see that we need to solve the following two systems of equations.
• Steps to finding an inverse matrix
Let A be a square matrix of order n.
1. Form the n [pic] 2n matrix [A[pic] I[pic]].
2. Transform this matrix into reduced row-echelon form.
3. If this new matrix is of the form [I[pic][pic]B], then A is invertible and B = A[pic].
Example 3. Find the inverse of
[pic]
III. The Inverse of a 2 [pic] 2 Matrix
• Formula for finding the inverse of a 2 [pic] 2 matrix:
[pic]
Example 4. Find the inverse of
[pic]
IV. Systems of Linear Equations
• If A is invertible, then system of equations represented by AX = B
has a unique solution X = A[pic]B.
Example 5. Solve the following system of equations.
[pic]
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