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Definition of The Matrix Identity
The identity matrix I n is the square matrix with order n x n and with the elements in the main diagonal consiting of 1's and all other elements are equal to zero.
Examples:
The 2 x 2 identity matrix
I 2 = [1 0]
[0 1]
The 3 x 3 identity matrix
[1 0 0]
[0 1 0]
[0 0 1]
In general the n x n matrix identity has the form
[1 0 ... 0]
[0 1 ... 0]
[0 0 ... 0]
A = [. ]
[. ]
[0 0 ... 1]
Definition of The Inverse of a Matrix
Let A be a square matrix of orde n x n. If there exists a matrix B such that
A B = I n = B A
Then B is called the inverse matrix of A. (and of course matrix A is the inverse matrix of B).
Examples:
Verify that matrices A and B given below are inverses of each other.
A = [2 3]
[3 4]
B = [-4 3]
[3 -2]
Solution
Let us find the products AB and BA
AB = [2 3][-4 3] = [-8 + 9 6 - 6] = [1 0]
[3 4][3 -2] [-12 + 12 9 - 8] [0 1]
BA = [-4 3][2 3] = [-8 + 9 -12 + 12] = [1 0]
[3 -2][3 4] [6 - 6 9 - 8] [0 1]
We have verified that AB = BA = I 2 and therefore A and B are inverse of each other.
The inverse of a square matrix A is denoted as A -1.
How to Find The Inverse of a 2 by 2 Square Matrix?
Example 1: Find a formula for the inverse of a 2 x 2 square matrix A given by
A = [a b]
[c d]
Solution
Let B be the inverse of A given by
B = [a' b']
[c' d']
From definition, we have
AB = I
[a b][a' b'] = [1 0]
[c d][c' d'] [0 1]
We need to find terms a', b', c' and d' of B (inverse) in terms of a, b, c and d which are the terms of matrix A. Multiply matrices A and B above.
[a a' + b c' a b' + b d']= [1 0]
[c a' + d c' c b' + d d'] [0 1]
The above gives 4 equations
a a' + b c' = 1 (equation 1)
c a' + d c' = 0 (equation 2)
a b' + b d' = 0 (equation 3)
c b' + d d' = 1 (equation 4)
We now need to solve the first and second equations simultaneously to find a' and c' in terms of a, b, c and d.
Multiply the first equation by d and the second equation by b and subtract the left and right terms of the equations obtained to find
a'(a d - b c) = d
a ' = d / (a d - b c)
Substitute a' in the second equation to obtain c'= - c / (a d - b c)
To find b' and d', multiply equation (3) by d and equation (4) by b and subtract the left and right terms of the equations obtained to find
b'(a d - b c) = - b
or b' = - b / (a d - b c)
Substitute b' in equation (3) to obtain d'= a / (a d - b c)
Finally matrix B , the inverse of A, is given by
B = 1 / (a d - b c) [d -b]
[-c a]
In solving the question in the above example, we have found a formula for the inverse of any invertible 2 x 2 matrix. A 2 x 2 matrix will have an inverse if it determinant D = a d - b c is not equal to zero since division by zero is not allowed.
Example 2: Use the above formula for the inverse of a 2 x 2 square matrix to find the inverse of
A = [3 5]
[2 3]
Solution
Acoording to formula above the inverse of A , A -1 is given by
A -1 = 1 / [ (3)(3) - (5)(2) ] [ 3 -5 ]
[-2 3]
Simplify to obtain
A -1 = [ -3 5 ]
[2 -3]
As an exercise, show that the product A A -1 = A -1 A = I 2.
How to Find The Inverse of a 3 by 3 Square Matrix?
Example 1: Find the inverse of a 3 x 3 square matrix A given by
[1 1 2]
A = [2 4 -3]
[3 6 -5]
Solution
We use a method based on augmenting the given matrix by the identity matrix as follows
[1 1 2 | 1 0 0] row(1)
[2 4 -3 | 0 1 0] row(2)
[3 6 -5 | 0 0 1] row(3)
We next use basic row operations so that the identity matrix on the right is moved to the left
We multiply row(1) by - 2 and add it to row(2) and put the result back in row(2). We also multiply row(1) by - 3 and add it to row(3) and put the result back in row(3).
[1 1 2 | 1 0 0] row(1)
[0 2 -7 | -2 1 0] row(2)
[0 3 -11 | -3 0 1] row(3)
We multiply row(2) by 3 and row(3) by -2, add them and put the row obtained in row(3).
[1 1 2 | 1 0 0] row(1)
[0 2 -7 | -2 1 0] row(2)
[0 0 1 | 0 3 -2] row(3)
We multiply row(1) by -2 and add it to row(2) and put the row obtained in row(1).
[-2 0 -11 | -4 1 0] row(1)
[0 2 -7 | -2 1 0] row(2)
[0 0 1 | 0 3 -2] row(3)
We multiply row(3) by 11 and add it to row(1) and put the row obtained in row(1).
[-2 0 0 | -4 34 -22] row(1)
[0 2 -7 | -2 1 0] row(2)
[0 0 1 | 0 3 -2] row(3)
We multiply row(3) by 7 and add it to row(2) and put the row obtained in row(2).
[-2 0 0 | -4 34 -22] row(1)
[0 2 0 | -2 22 -14] row(2)
[0 0 1 | 0 3 -2] row(3)
We now multiply all terms in row(1) by -1/2 and multiply all terms in row(2) by 1/2.
[1 0 0 | 2 -17 11] row(1)
[0 1 0 | -1 11 -7] row(2)
[0 0 1 | 0 3 -2] row(3)
Now that we have the identity matrix on the left, the 3 by 3 matrix on the right is the inverse matrix. The inverse of A is given by
[2 -17 11]
A -1 = [-1 11 -7]
[0 3 -2]
A similar method based on row operations can in principle be used to find the inverse of any invertible (that has an inverse) square matrix.
Exercises
Find the inverse of matrices A and B given below.
A = [2 3]
[3 5]
[1 1 2]
B = [1 2 4]
[2 -1 0]
Answers
A -1 = [5 -3]
[-3 2]
[2 -1 0]
B -1 = [4 -2 -1]
[-5/2 3/2 1/2]
More pages and references related to matrices.
Find Inverse Matrix - Calculator.
Find Inverse of 3 by 3 Matrix - Calculator.
Matrix Addition and Multiplication.
multiplication of matrices using an applet.
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