Chapter 5 Determinants

Transcription

1 hpter 5 Determinnts

2 5. Introduction Every squre mtri hs ssocited with it sclr clled its determinnt. Given mtri, we use det() or to designte its determinnt. We cn lso designte the determinnt of mtri by replcing the brckets by verticl stright lines. For emple, det( ) Definition : The determinnt of mtri [] is the sclr. Definition : The determinnt of mtri c b d is the sclr d-bc. For higher order mtrices, we will use recursive procedure to compute determinnts.

3 5. Epnsion by ofctors Definition : Given mtri, minor is the determinnt of ny squre submtri of. Definition : Given mtri =[ ij ], the cofctor of the element ij is sclr obtined by multiplying together the term (-) i+j nd the minor obtined from by removing the ith row nd the jth column. In other words, the cofctor ij is given by ij = () i+j M ij. For emple, M M ( ) M M ( ) M M

4 5. Epnsion by ofctors To find the determinnt of mtri of rbitrry order, Pick ny one row or ny one column of the mtri; For ech element in the row or column chosen, find its cofctor; Multiply ech element in the row or column chosen by its cofctor nd sum the results. This sum is the determinnt of the mtri. In other words, the determinnt of is given by det( ) n j ij ij i i i i in in ith row epnsion det( ) n i ij ij j j j j nj nj jth column epnsion

5 Emple : We cn compute the determinnt T by epnding long the first row, T Or epnd down the second column: 6 T Emple : (using row or column with mny zeroes) 5 5 6

6 5. Properties of determinnts Property : If one row of mtri consists entirely of zeros, then the determinnt is zero. Property : If two rows of mtri re interchnged, the determinnt chnges sign. Property : If two rows of mtri re identicl, the determinnt is zero. Property : If the mtri B is obtined from the mtri by multiplying every element in one row of by the sclr λ, then B = λ. Property 5: For n n n mtri nd ny sclr λ, det(λ)= λ n det().

7 5. Properties of determinnts Property 6: If mtri B is obtined from mtri by dding to one row of, sclr times nother row of, then = B. Property : det() = det( T ). Property 8: The determinnt of tringulr mtri, either upper or lower, is the product of the elements on the min digonl. Property 9: If nd B re of the sme order, then det(b)=det() det(b).

8 5. Pivotl condenstion Properties,, 6 of the previous section describe the effects on the determinnt when pplying row opertions. These properties comprise prt of n efficient lgorithm for computing determinnts, technique known s pivotl condenstion. - given mtri is trnsformed into row-reduced form using elementry row opertions - record is kept of the chnges to the determinnt s result of properties,, 6. -Once the trnsformtion is complete, the row-reduced mtri is in upper tringulr form, nd its determinnt is esily found by property 8. Emple in the net slide

9 Find the determinnt of () Fctor out of the nd row ( ) ) ()()( 5. Pivotl condenstion

10 5.5 Inversion Theorem : squre mtri hs n inverse if nd only if its determinnt is not zero. Below we develop method to clculte the inverse of nonsingulr mtrices using determinnts. Definition : The cofctor mtri ssocited with n n n mtri is n n n mtri c obtined from by replcing ech element of by its cofctor. Definition : The djugte of n n n mtri is the trnspose of the cofctor mtri of : = ( c ) T

11 Find the djugte of Solution: The cofctor mtri of : 6 6 Emple of finding djugte

12 Inversion using determinnts Theorem : = = I. If then from Theorem, if I Tht is, if, then - my be obtined by dividing the djugte of by the determinnt of. For emple, if then, d c b c b d bc d

13 Use the djugte of to find - 6 )() ()( ()()() ) )( )( ( 6 Inversion using determinnts: emple

14 5.6 rmer s rule If system of n liner equtions in n vribles =b hs coefficient mtri with nonzero determinnt, then the solution of the system is given by det( ) det( ) det( ),,,, det( ) det( ) n n det( ) where i is mtri obtined from by replcing the ith column of by the vector b. Emple: b b b b b b

15 Use rmer s Rule to solve the system of liner eqution. z y z z y 5 8 )()() ( ()()() 5 8, z y 5.6 rmer s rule: emple