Chapter 04.04 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 25 20 5 0 6 6 5 7 2 25 27 he traspose of is 25 5 6 20 0 6 A 5 7 2 25 27 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. 04.04.
04.04.2 Chapter 04.04 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. 2.2.2 6.2 2.5 6 9. is a symmetric matrix as a a. 2, a a 6 ad a a. What is a skew-symmetric matrix? A 2 2 2 2 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. 0 2 0 5 2 5 0 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 5 2 6 4 2 7 2 6
Uary atrix Operatios 04.04. 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. 25 20 2 5 0 5 25 6 6 7 27 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..25 0.0 5.02 0.05 [B] 40.9.02 4.0.2 25.0.02 27.0.95 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].25 0.0 5.02 0.05 40.9.02 4.0.2 25.0.02 27.0.95.25 40.9 25.0 0.0.02.02 [C] 5.02 4.0 27.0 0.05.2.95 Recogize ow that if we fid [ A ][ C], we get D AC
04.04.4 Chapter 04.04.25 40.9 25.0 25 20 2 0.0.02.02 5 0 5 25 5.02 4.0 27.0 6 6 7 27 0.05.2.95 597 965 9 74 252 25 76 269 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 597 252 $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,,
Uary atrix Operatios 04.04.5 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 25 5 44 2 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 25 44 a2 2 a 5 2 2 4 25 5 44 2
04.04.6 Chapter 04.04 44 0 25 44 5 2 44 2 4 ( A) a a2 2 a 25 4 5 0 4 00 400 4 4 Also for i, ethod 2: A j a jc C 4 2 C 2 2 0 C 2 4 A a C i j a2c2 ac ( 25) 4 (5) 0 () 4 i 00 400 4 4 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
Uary atrix Operatios 04.04.7 2 2 25 44 44 0 25 44 5 2 5 2 25 44 9 25 5 44 2 25 9 ( A) a2 2 a a2 2 5( 0) ( 9) 2( 9) 400 952 46 4 I terms of cofactors for j 2, A a i C i 2 i 2 2 C 2 2 0 C 9 2 C 2 9 A a 2 2 2 2C2 ac a2c2 ( 5) 0 () 9 (2) 9 400 952 46
04.04. Chapter 04.04 4 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 0 2 6 0 7 4 0 4 9 5 0 5 2 Sice oe of the colums (first colum i the above example) of is a zero, ( A ) 0. Example What is the ermiat of 2 6 4 2 7 6 5 4 2 0 9 5 (A ) is zero because the fourth colum i
Uary atrix Operatios 04.04.9 4 6 0 is 2 times the first colum 2 5 9 Example 9 If the ermiat of 25 5 44 2 is 4, the what is the ermiat of 25 0.5 [B] 6. 44 25.2 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 25 5 44 2 is 4, what is the ermiat of 25 5 [B] 0 4..56 44 2
04.04.0 Chapter 04.04 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 25 5 0 4..56 0 0 0.7 Sice is a upper triagular matrix A i a ii a a a 25 4.0.7 4 Key erms: raspose Symmetric atrix Skew-Symmetric atrix race of atrix Determiat