Matrices : Cayley Hamilton Theorem I :Best Engineering Mathematics Tips (AU,JNTU,GATE,DU)

now I am going to teach you the Callie Hamilton theorem cayley-hamilton theorem what is Cal a Hamilton theorem every square matrix should satisfy its own characteristic equation the work satisfies is very very important even for too much they will ask you the statement of Calais Hamilton theorem so what is it every square matrix every square matrix satisfies it's own characteristic equation now I will show you some example find the or verify they'll give you some matrix and ask you to verify whether the given matrix is Cali Hamilton theorem or not and also they will ask you to find the inverse of the given matrix in the Cal a Hamilton theorem they will first ask you to prove whether the given matrix is Cali Hamilton or not and then hence find the inverse of the given matrix so how will you find it what is the working rule to find the Cali Hamilton theorem first you need to find the characteristic equation after finding out the characteristic equation you will replace lambda by a wherever there is lambda you will replace it by a here no need to find out for any eigen values or any eigenvectors the only thing you need to know is matrix multiplication you have to be thorough with the matrix multiplication when two matrices are given of the same order then you have to go by rows into columns rows into columns this is all you need to find the matrix multiplication so here no need to find out any eigen values or eigenvectors after finding out the characteristic equation straight away you can substitute replace lambda by a and substitute the given matrix and a square a cube matrices now find the out verify Cali Hamilton theorem verify cayley-hamilton theorem and hence find the inverse and hence find the inverse of the matrix is equals to 1 2 minus 1 3 minus 3 1 2 1 minus 2 so this is how they will give you the question verify Calley Hamilton theorem and hence the fine inverse of the matrix now to verify the Calley Hamilton theorem what is the condition you need to get 0 matrix that means the characteristic equation should satisfy its own characteristic equation that is every square matrix so first to find the verify the Calla Hamilton theorem is for 8 marks and to find the inverse of the matrix is for 8 marks so totally 16 mark question is this so first thing you will take lit a be the given matrix then find out characteristic equation to find the characteristic equation what is the formula lambda cube minus s1 lambda square plus s 2 lambda minus s 3 equal to 0 as I told you in the previous topic s1 is the sum of the diagonal elements yes 2 is the sum of the minus of the diagonal elements and the s3 is the determinant of a now first find out what is s 1 s 1 is equals to 1 minus 3 minus 2 1 minus 3 minus 2 which will give you minus 4 similarly s 2 will be the sum of the minus of the diagonal elements that is when you take this is the diagonal element when you take the first element what is the minus here minus 3 1 1 minus 2 plus 4 the next diagonal element this is the next diagonal element so what is left out 1 minus 1/2 plus 2 1 minus 1 2 minus 2 plus 4 the last one for this one 1 2 3 minus 3 so when you simplify all these you will get the value as minus 4 now we need to find the determinant of a that is s 3 how will you find out s 3 as I told you earlier 1 into 6 minus 1 minus 2 alternative signs plus minus plus minus minus 2 into again minus 6 minus 2 and then minus 1 into 3 plus 6 which will give you 5 minus into minus will become plus 16 minus 9 which is equals to 12 therefore we have got s 1 s 2 s 3 now write down the characteristic equation therefore the characteristic equation is lambda cube minus s 1 already minus sign is there so minus into minus will become plus 4 lambda square and then plus s 2 so minus 4 lambda and minus 12 equal to 0 is the characteristic equation is the characteristic equation now once you have done with the characteristic equation you have to check it once or twice because if even the single value or even the signs plus or minus sign changes the final answer you will not get it as zero matrix okay to get the zero matrix proper answer you need to check the characteristic equation which is very very important for this topic now after finding out the characteristic equation you are no need to find out any eigen values or eigenvectors straight away I am going to replace lambda by ay so what do you get a Q plus 4 a square minus 4 a minus 12 into this will become a unit matrix which is equal to 0 so here no need to find out anything any a formula or are no applications nothing just the matrix multiplication a cube a square and the a is a given matrix I is the unit matrix first we need to find out a square matrix what is a square a into 8 that is the given matrix a into 8 1 2 minus 1 3 minus 3 1 and then 2 1 minus 2 into the same thing 1 2 minus 1 3 minus 3 1 and then 2 1 minus 2 these are the matrices a into 8 now I am going to multiply these two rows into columns how will you multiply first 1 into 1 plus 2 into 3 6 1 plus 6 will give you 7 plus minus 1 into 2 I'll show you 1 plus 2 into 3 6 minus 1 into 2 minus 2 that is for the first element next one again the same row into next column that is 2 1 2 2 and then 2 into minus 3 which will give you minus 6 again 2 into 1 which will give you plus 2 this is for the second element similarly the same row into third column which will give you minus 1 plus 2 plus 4 this is how you need to find the all the elements that is for the second one similarly how will you do it you will get it as 3 into 1 and then minus 9 plus 2 next one the same second row into the second column 6 plus 9 plus 1 and then the same row into the third column this is how you need to find the matrix a square I will just write away write down the matrix here so that we are one I hope you have understood this one just the rows into columns this is all you need to find the matrix a square that is 5 minus 5 3 minus 4 16 minus 8 and 1 minus 1 3 so after finding out the a square matrix check it once again because if the any of the value in the square matrix a square matrix is wrong then you will not get the a cube matrix and finally you will not get the a cube final answer zero matrix now how will you find out the a cube a cube is equals to you have already got the matrix a square so a square into a will give you the a cube matrix so what is the a square matrix 5 - 5 3 - 4 16 - 8 and then 1 - 1 3 into the given matrix 1 2 minus 1 3 minus 3 1 - 1 - 2 now same way the matrix multiplication rows into column for the first element again the same row in to second column same row into third column will give you the first row elements so you know how to do with the a cube matrix or directly I will write down - 4 28 - 16 and then 28 - 64 36 and then 4 8 - 8 this is what the a cube matrix now what is the next step take this equation as someone or something take it as some equation 1 substituting substitute the value of a cube a square yay and I in the characteristic equation in equation 1 so what do you get when you substitute this a cube matrix minus 4 28 minus 16 28 minus 64 36 4 8 minus 8 we have substituted a cube matrix plus 4 into what is the a square matrix 5 minus 5 3 minus 4 16 - 8 1 minus 1 3 a square matrix next one matrix a 4 into 1 2 minus 1 3 minus 3 1 2 1 minus 2 and finally minus 12 into I what is I the identity matrix that is 1 0 0 0 1 0 0 0 1 so when you simplify all these how will you simplify minus 4 plus 4 into 5 20 minus 4 into 1 4 minus 12 you will get it as 0 similarly you will get 0 0 0 you will get the matrix as 0 therefore we say that the callee Hamilton theorem is verified so what is our aim we need to find the zero matrix once you find out the characteristic equation you will replace lambda by a throughout the equation after replacing then you will find out a cube matrix a square a matrix finally after finding out by using the matrix multiplication you will substitute in the characteristic equation final answer you have to get it as zero matrix this is oh you will verify Callie Hamilton theorem

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Id: u5hrdFsIql8

Length: 15min 54sec (954 seconds)

Published: Mon Oct 03 2016