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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