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 Rn to Rm 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  =  In   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.  If A is nonsingular, then so is A-1 and           (A-1) -1  =  A If A and B are nonsingular matrices, then AB is nonsingular and         (AB) -1  =  B-1A-1 -1 If A is nonsingular then           (AT) -1  =  (A -1)T If A and B are matrices with         AB  =  In then A and B are inverses of each other. 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 jth column of the product by multiplying A by the jth column of B.  Now for some notation.  Let ej be the m x 1 matrix that is the jth column of the identity matrix and xj be the jth column of B.  Then          Axj  =  ej  We can write this in augmented form         [A|ej] 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-1Ax  =  A-1b         Ix  =  A-1b        x  =  A-1b Example Solve         x + 4z  =  2         x + y + 6z  =  3         -3x - 10z  =  4 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-10  =  0 This demonstrates a theorem Theorem of Nonsingular Equivalences The Following Are Equivalent (TFAE) A is nonsingular Ax  =  0 has only the trivial solution A is row equivalent to I The linear system Ax  =  b has a unique solution for every n x 1 matrix b Back to the Linear Algebra Home Page