Matrix II I.

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Matrix II

I. Key mathematical terms

Terms Symbol Chinese translation

Reciprocal

Inverse Matrix

Transpose

II. Inverse Matrix

I

n real numbers, ifa0there exists a unique reciprocalb ¹

 a which satisfies

"abba1".

Definition:(Inverse Matrix)

Let A be a n×n square matrix. We say A is invertible (or nonsingular) if there exists a n×n square matrix B such that

ABBAIn

If such matrix B exists, we say B is” the inverse of A” and denoted asBA^¹. If no such inverse B exists for a matrix A, we say A is non-invertible (or singular).

<key> There can only exist at most one inverse.

<key>The matrix is singular if and only if det(A) = 0.

Example1.

3 2 2 1

1 1 , 2 1

A   B  

   

For each of the matrices A and B determine whether the matrix is singular. If the matrix is non-singular, find its inverse.

(Hint: You can use Cramer’s rule to solve the linear equation.)

(1) ³ ²

A  1 1

  

(2) ^{2 1}

B 2 1

  

 

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Inverse of 2x2 Matrix

W

e can fine the inverse of any non-singular matrix.

The inverse of a matrix M is the matrixM^¹such thatMM^¹M M^¹ I . In the case of 2x2 matrix A ^a ^b

c d

 

  

  a simple formula exists to find its inverse:

If A ^a ^b c d

 

  

  then ¹ ¹ ¹

det( )

d b d b

A ad bc c a A c a

        

<key>

(1) If det(A)=0, then A is noninvertible.

(2) If det(A)≠0, then A is invertible.

<proof>

To varify the formula above, suppose B ^x ^u y v

 

  

  such thatABBAI₂.

2

1 0

= 0 1 AB I

a b x u c d y v



    

     

    

We can get two systems of linear equations.

1 0

and

0 1

ax by au bv

cx dy cu dv

     

       We can use Cramer’s rule to solve these

equations.

(1) ¹

0 ax by cx dy

 

  



(2) ⁰

1 au bv

cu dv

 

  



Finally we have:

ForA ^a ^b

c d

 

  

 

1 1 1

A^     

   

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Example2.

5 3 7 8

Given ,

2 1 4 5

A  B   (1) Find the inverse matrix of A.

(2) If AX=AB, Find X

III. Solving systems of equations by using matrices

We can use the inverse of n×n matrix to solve a system of n simultaneous linear equations in n unknowns. We will introduce how to solve the 2 simultaneous linear equations in 2 unknowns in our class.

If x and is non-singular, then x 1 .

A v A A v

y y

    

   

   

Example3.

Use an inverse matrix to solve the simultaneous equations: ² ⁴ ⁴

3 5 3

x y x y

 

  



Example4.

Matrix A is a 2x2 matrix which satisfies ⁸ ¹ , ³ ¹

5 2 2 1

A       A       

        (1) Find matrix A

(2) If ³ ⁰ , 0 3 AB  

  

  find B

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IV. Properties of Inverse and Transpose Definition:(Transpose of Matrix)

G

iven an m×n matrix W we define W^t, the transpose of W, to be the n×m matrix whose (i,j)th entry is the (j,i)th entry of W ; that is the matrix for which

(W^t)_ij W_ji For example

1 2

1 0 4

0 1 ;

2 1 3

4 3

t a b t a c

A A B B

c d b d

 

        

         

<key> Transpose of matrix is found by interchanging rows into columns (or columns into rows).

Properties of Inverse and Transpose

(1) A martix has at most one inverse.

(2) Assume A and B are invertible n×n matrices. Then the product AB is invertible and

1 1 1

(AB)^ B A^ ^

(3) We have (AB)^t B A^t ^t and (A)^t A^t for any matrix A and real constant .

(4) Assume A is invertible. Then the transpose A^t is invertible and

1 1

(A^t)^ (A^ )^t

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(2)

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(Hint: Use the prorperties of inverse and transpose to solve example5~7.) Example5.

Matrix A is a 2x2 matrix and ² ³ ⁴ , ³ ^{11 15}

8 11 30 41

A   A  

    Find A.

Example6.

Matrix A and B are invertible 2x2 matrices. Suppose ³ ⁴

5 7

AB   

  . Find BA.

Example7.

Matrix A and B are invertible 2x2 matrices, such that BAB=I.

(1) Prove thatAB B^¹ ^¹. Given that ² ⁵ 1 3

B  

  

  (2) Find the matrix A such that BAB=I

製作者：國立臺灣師範大學附屬高級中學蕭煜修