Inverse of a Matrix Definition and Examples Recall that functions f and g are inverses if f(g(x)) = g(f(x)) = x We will see later that matrices can be considered as functions from R^{n} to R^{m} and that matrix multiplication is composition of these functions. With this knowledge, we have the following: Let A and B be n x n matrices then A and B are inverses of each other, then AB = BA = I_{n}
Example Consider the matrices
We can check that when we multiply A and B in either order we get the identity matrix. (Check this.) Not all square matrices have inverses. If a matrix has an inverse, we call it nonsingular or invertible. Otherwise it is called singular. We will see in the next section how to determine if a matrix is singular or nonsingular. Properties of Inverses Below are four properties of inverses.
Notice that the fourth property implies that if AB = I then BA = I. The first three properties' proof are elementary, while the fourth is too advanced for this discussion. We will prove the second. Proof that (AB) ^{1} = B^{ 1} A^{ 1} By property 4, we only need to show that (AB)(B ^{1} A ^{1}) = I We have (AB)(B ^{1} A ^{1}) = A(BB ^{1})A ^{1} associative property = AIA^{1 } definition of inverse = AA^{1 } definition of the identity matrix = I definition of inverse Finding the Inverse Now that we understand what an inverse is, we would like to find a way to calculate and inverse of a nonsingular matrix. We use the definitions of the inverse and matrix multiplication. Let A be a nonsingular matrix and B be its inverse. Then AB = I Recall that we find the j^{th} column of the product by multiplying A by the j^{th} column of B. Now for some notation. Let e_{j} be the m x 1 matrix that is the j^{th} column of the identity matrix and x_{j} be the j^{th} column of B. Then Ax_{j} = e_{j} We can write this in augmented form [Ae_{j}] Instead of solving these augmented problems one at a time using row operations, we can solve them simultaneously. We solve [A  I]
Example Find the inverse of the matrix
Solution
The inverse matrix is just the right hand side of the final augmented matrix
This example demonstrates that if A is row equivalent to the identity matrix then A is nonsingular. Linear Systems and Inverses We can use the inverse of a matrix to solve linear systems. Suppose that Ax = b Then just as we divide by a coefficient to isolate x, we can apply A^{1} to both sides to isolate the x. A^{1}Ax = A^{1}b Ix = A^{1}b x = A^{1}b
Example Solve
x + 4z = 2 Solution We put this system in matrix form Ax = b with
The solution is x = A^{1} b We have already computed the inverse. We arrive at
The solution is x = 18 y = 9 z = 5 Notice that if b is the zero vector, then Ax = 0 can be solved by x = A^{1}0 = 0 This demonstrates a theorem Theorem of Nonsingular Equivalences The Following Are Equivalent (TFAE)
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