How to Find the Inverse for Any Square Matrix
In this lesson, we describe a method for finding the inverse of any square matrix; and we demonstrate the method step-by-step with
examples.
Prerequisites: This material assumes familiarity with elementary matrix operations and echelon transformations.
How to Find the Inverse of an n x n Matrix
Let A be an n x n matrix. To find the inverse of matrix A, we follow these steps:
1. Using elementary operators, transform matrix A to its reduced row echelon form, Arref.
2. Inspect Arref to determine if matrix A has an inverse.
If Arref is equal to the identity matrix, then matrix A is full rank; and matrix A has an inverse.
If the last row of Arref is all zeros, then matrix A is not full rank; and matrix A does not have an inverse.
3. If A is full rank, then the inverse of matrix A is equal to the product of the elementary operators that produced Arref , as shown
below.
A-1 = Er Er-1 . . . E2 E1
where
A-1 = inverse of matrix A
r = Number of elementary row operations required to transform A to Arref
Ei = ith elementary row operator used to transform A to Arref
Note that the order in which elementary row operators are multiplied is important, because Ei Ej is not necessarily equal to Ej Ei.
An Example of Finding the Inverse
Let's use the above method to find the inverse of matrix A, shown below.
1 2 2
A = 2 2 2
2 2 1
The first step is to transform matrix A into its reduced row echelon form, Arref, using a series of elementary row operators Ei. We show
the transformation steps below for each elementary row operator.
1. Multiply row 1 of A by -2 and add the result to row 2 of A. This can be accomplished by pre-multiplying A by the elementary ro
operator E1, which produces A1.
1 0 0 1 2 2
E1 = -2 1 0 A1 = E 1 A = 0 -2 -2
0 0 1 2 2 1
In this lesson, we describe a method for finding the inverse of any square matrix; and we demonstrate the method step-by-step with
examples.
Prerequisites: This material assumes familiarity with elementary matrix operations and echelon transformations.
How to Find the Inverse of an n x n Matrix
Let A be an n x n matrix. To find the inverse of matrix A, we follow these steps:
1. Using elementary operators, transform matrix A to its reduced row echelon form, Arref.
2. Inspect Arref to determine if matrix A has an inverse.
If Arref is equal to the identity matrix, then matrix A is full rank; and matrix A has an inverse.
If the last row of Arref is all zeros, then matrix A is not full rank; and matrix A does not have an inverse.
3. If A is full rank, then the inverse of matrix A is equal to the product of the elementary operators that produced Arref , as shown
below.
A-1 = Er Er-1 . . . E2 E1
where
A-1 = inverse of matrix A
r = Number of elementary row operations required to transform A to Arref
Ei = ith elementary row operator used to transform A to Arref
Note that the order in which elementary row operators are multiplied is important, because Ei Ej is not necessarily equal to Ej Ei.
An Example of Finding the Inverse
Let's use the above method to find the inverse of matrix A, shown below.
1 2 2
A = 2 2 2
2 2 1
The first step is to transform matrix A into its reduced row echelon form, Arref, using a series of elementary row operators Ei. We show
the transformation steps below for each elementary row operator.
1. Multiply row 1 of A by -2 and add the result to row 2 of A. This can be accomplished by pre-multiplying A by the elementary ro
operator E1, which produces A1.
1 0 0 1 2 2
E1 = -2 1 0 A1 = E 1 A = 0 -2 -2
0 0 1 2 2 1
