Chapter Unary Matrix Operations

Transcription

1 Chapter Uary atrix Operatios After readig this chapter, you should be able to:. kow what uary operatios meas, 2. fid the traspose of a square matrix ad it s relatioship to symmetric matrices,. fid the trace of a matrix, ad 4. fid the ermiat of a matrix by the cofactor method. What is the traspose of a matrix? Let be a m matrix. he [B] is the traspose of the if b a for all i ad j. hat is, the i th row ad the j th colum elemet of is the j th row ad i th colum elemet of [B]. Note, [B] would be a m matrix. he traspose of is deoted by. Example Fid the traspose of he traspose of is A Note, the traspose of a row vector is a colum vector ad the traspose of a colum vector is a row vector. Also, ote that the traspose of a traspose of a matrix is the matrix itself, that is, A A. Also, A B A B ; ca ca

2 Chapter What is a symmetric matrix? A square matrix with real elemets where a a ji for i,2,..., ad j,2,..., is called a symmetric matrix. his is same as, if [ A] [ A], the is a symmetric matrix. Example 2 Give a example of a symmetric matrix is a symmetric matrix as a a. 2, a a 6 ad a a. What is a skew-symmetric matrix? A matrix is skew symmetric if a a ji for i,..., ad j,...,. his is same as A A. Example Give a example of a skew-symmetric matrix is skew-symmetric as a 2 a2 ; a a 2; a2 a2 5. Sice aii aii oly if a ii 0, all the diagoal elemets of a skew-symmetric matrix have to be zero. What is the trace of a matrix? he trace of a tr A matrix is the sum of the diagoal etries of, that is, a ii i Example 4 Fid the trace of

3 Uary atrix Operatios tr A Example 5 a ii i ( 5) ( 4) (6) 7 he sales of tires are give by make (rows) ad quarters (colums) for Blowout r us store locatio A, as show below where the rows represet the sale of irestoe, ichiga ad Copper tires, ad the colums represet the quarter umber, 2,, 4. Fid the total yearly reveue of store A if the prices of tires vary by quarters as follows [B] where the rows represet the cost of each tire made by irestoe, ichiga ad Copper, ad the colums represet the quarter umbers. o fid the total tire sales of store A for the whole year, we eed to fid the sales of each brad of tire for the whole year ad the add to fid the total sales. o do so, we eed to rewrite the price matrix so that the quarters are i rows ad the brad ames are i the colums, that is, fid the traspose of [B]. [ C] [ B] [C] Recogize ow that if we fid [ A ][ C], we get D AC

4 Chapter he diagoal elemets give the sales of each brad of tire for the whole year, that is d $597 (irestoe sales) d $252 (ichiga sales) d $ (Cooper sales) he total yearly sales of all three brads of tires are i d ii $5060 ad this is the trace of the matrix. Defie the ermiat of a matrix. he ermiat of a square matrix is a sigle uique real umber correspodig to a matrix. For a matrix, ermiat is deoted by A or (A ). So do ot use ad A iterchageably. For a matrix, a a2 [ A ] a2 a ( A) a a a a 2 2 How does oe calculate the ermiat of ay square matrix? Let be matrix. he mior of etry a is deoted by ad is defied as the ermiat of the ( ( ) submatrix of, where the submatrix is obtaied by deletig the i th or row ad i A j th j colum of the matrix. he ermiat is the give by j a for ay i, 2,, i A i j a for ay j, 2,,

5 Uary atrix Operatios coupled with that A a for a matrix [ A] of a matrix to ermiats of matrices. he umber, as we ca always reduce the ermiat i j ( ) is called the cofactor of a ad is deoted by c. he above equatio for the ermiat ca the be writte as or A a C for ay i, 2,, A j i a C for ay j, 2,, he oly reaso why ermiats are ot geerally calculated usig this method is that it becomes computatioally itesive. For a matrix, it requires arithmetic operatios proportioal to!. Example 6 Fid the ermiat of ethod : i A j Let i i the formula A 2 j j j a a j j for ay i, 2, 2 a a2 2 a a a2 2 a

6 Chapter ( A) a a2 2 a Also for i, ethod 2: A j a jc C 4 2 C C 2 4 A a C i j a2c2 ac ( 25) 4 (5) 0 () 4 i j A a for ay j,2, Let 2 j i the formula i A i 2 a i2 i2 2 2 a a a2 2 a 2 2 a a2 2

7 Uary atrix Operatios ( A) a2 2 a a2 2 5( 0) ( 9) 2( 9) I terms of cofactors for j 2, A a i C i 2 i 2 2 C C 9 2 C 2 9 A a C2 ac a2c2 ( 5) 0 () 9 (2)

8 Chapter Is there a relatioship betwee (AB), ad (A) ad (B)? Yes, if ad [B] are square matrices of same size, the ( AB) ( A)( B) Are there some other theorems that are importat i fidig the ermiat of a square matrix? heorem : If a row or a colum i a matrix is zero, the ( A ) 0. heorem 2: Let be a matrix. If a row is proportioal to aother row, the ( A ) 0. heorem : Let be a matrix. If a colum is proportioal to aother colum, the ( A ) 0. heorem 4: Let be a matrix. If a colum or row is multiplied by k to result i matrix k, the ( B) k ( A). heorem 5: Let be a upper or lower triagular matrix, the ( B) aii. Example 7 What is the ermiat of Sice oe of the colums (first colum i the above example) of is a zero, ( A ) 0. Example What is the ermiat of (A ) is zero because the fourth colum i

9 Uary atrix Operatios is 2 times the first colum Example 9 If the ermiat of is 4, the what is the ermiat of [B] Sice the secod colum of [B] is 2. times the secod colum of ( B) 2.( A) ( 2.)( 4) 76.4 Example 0 Give the ermiat of is 4, what is the ermiat of 25 5 [B]

10 Chapter Sice [B] is simply obtaied by subtractig the secod row of by 2.56 times the first row of, (B) (A) 4 Example What is the ermiat of Sice is a upper triagular matrix A i a ii a a a Key erms: raspose Symmetric atrix Skew-Symmetric atrix race of atrix Determiat