Home Topics in Analysis Analysis Elementary Linear Algebra Linear Algebra Analysis of Functions of Many Variables Linear Algebra and Analysis Complex Analysis Calculus of One And Many Variables Engineering Math One Variable Advanced Calculus Interrogation of Hiroshi Oshima

82 CHAPTER 4. MATRICES

4.2 Finding the Inverse of a MatrixLater a formula is given for the inverse of a matirx. However, it is not a good way to findthe inverse for a matrix. There is a much easier way and it is this which is presented here.It is also important to note that not all matrices have inverses.

Example 4.2.1 Let A =

(1 11 1

). Does A have an inverse?

One might think A would have an inverse because it does not equal zero. However,(1 11 1

)(−11

)=

(00

)

and if A−1 existed, this could not happen because you could multiply on the left by theinverse A and conclude the vector (−1,1)T = (0,0)T . Thus the answer is that A does nothave an inverse.

Suppose you want to find B such that AB = I. Let

B =(

b1 · · · bn

)Also the ith column of I is

ei =(

0 · · · 0 1 0 · · · 0)T

Thus, if AB = I, bi, the ith column of B must satisfy the equation Abi = ei. The augmentedmatrix for finding bi is (A|ei) . Thus, by doing row operations till A becomes I, you end upwith (I|bi) where bi is the solution to Abi = ei. Now the same sequence of row operationsworks regardless of the right side of the agumented matrix (A|ei) and so you can savetrouble by simply doing the following.

(A|I) row operations→ (I|B)

and the ith column of B is bi, the solution to Abi = ei. Thus AB = I.This is the reason for the following simple procedure for finding the inverse of a matrix.

This procedure is called the Gauss Jordan procedure. It produces the inverse if the matrixhas one. Actually, it produces the right inverse.

Procedure 4.2.2 Suppose A is an n×n matrix. To find A−1 if it exists, form the augmentedn× 2n matrix, (A|I) and then do row operations until you obtain an n× 2n matrix of theform

(I|B) (4.14)

if possible. When this has been done, B = A−1. The matrix A has an inverse exactly whenit is possible to do row operations and end up with one like 4.14.

Here is a fundamental theorem which describes when a matrix has an inverse.

Theorem 4.2.3 Let A be an n× n matrix. Then A−1 exists if and only if the columns of Aare a linearly independent set. Also, if A has a right inverse, then it has an inverse whichequals the right inverse.

82 CHAPTER 4. MATRICES4.2 Finding the Inverse of a MatrixLater a formula is given for the inverse of a matirx. However, it is not a good way to findthe inverse for a matrix. There is a much easier way and it is this which is presented here.It is also important to note that not all matrices have inverses.1Example 4.2.1 Let A = ( '. Does A have an inverse?One might think A would have an inverse because it does not equal zero. However,CGC)and if A~! existed, this could not happen because you could multiply on the left by theinverse A and conclude the vector (—1,1)’ = (0,0)". Thus the answer is that A does nothave an inverse.Suppose you want to find B such that AB = J. LetB=(b, ve bn )Also the i” column of J isTei=(0 0 1 0: 0)Thus, if AB = /, b;, the /” column of B must satisfy the equation Ab; = e;. The augmentedmatrix for finding b; is (Ale;). Thus, by doing row operations till A becomes J, you end upwith (/|b;) where 6; is the solution to Ab; = e;. Now the same sequence of row operationsworks regardless of the right side of the agumented matrix (Ale;) and so you can savetrouble by simply doing the following.) row operations ((All |B)and the i” column of B is b;, the solution to Ab; = e;. Thus AB = /.This is the reason for the following simple procedure for finding the inverse of a matrix.This procedure is called the Gauss Jordan procedure. It produces the inverse if the matrixhas one. Actually, it produces the right inverse.Procedure 4.2.2 Suppose A is ann xn matrix. To find A~' if it exists, form the augmentedn X 2n matrix, (A\I) and then do row operations until you obtain an n x 2n matrix of theform(I|B) (4.14)if possible. When this has been done, B = A~'. The matrix A has an inverse exactly whenit is possible to do row operations and end up with one like 4.14.Here is a fundamental theorem which describes when a matrix has an inverse.Theorem 4.2.3 Let A be ann xn matrix. Then A~! exists if and only if the columns of Aare a linearly independent set. Also, if A has a right inverse, then it has an inverse whichequals the right inverse.