14 Linear Algebra Part 14 Cayley Hamilton theorem

追加: 2026-04-29

[00:00:00]so our first problem on inverse of a matrix is if ABCD are for invertible matrices invertible matrix implements non singular matrix determinant of all these four matrices is nonzero so if ABCD are four invertible matrices such that a into such that the product of these four matrices is identity matrix then wasan is we have to find B inverse now see three types of questions are generally asked on this concept suppose if this relation is given ABCD this equals to identity matrix and here if I ask you what is the inverse of matrix a so you can see that one matrix is a another matrix is BCD and the product of these two matrices is identity matrix so I can say that the second matrix is inverse of first matrix so this means inverse of matrix a this will be simply BCD so here you don't have to do any calculation you can directly say that a inverse is BCD the second type of problem that is asked is find inverse of matrix 8d so now you can see that one matrix is ABC another matrix is d and the product of these two matrices a B C into D this is identity matrix so this means a d inverse this will be ABC so if these type of problems are asked either a inverse then you can directly write BCD if the question is find D inverse then you can directly write ABC now here the question is we have to find the inverse of this matrix B see in case of matrix you cannot change the order we know that matrix multiplication is not commutative so you cannot write a be equals to B so here if I if I if I have to find inverse of matrix B then in that case your first M will be you have to separate this B so see how I'm going to separate this matrix B the relation that I have is ABCD this equals to identity matrix so we are since we have to separate this B so I am going to multiply both side by a inverse and this will be pre multiplication so first I am going to multiply this side by a inverse and on right hand side also first we have to multiply this by a inverse so this is the relation that you have now multiply both side by a inverse now a into a inverse this will be identity matrix so you will get BCD this equals to a inverse into identity so this will be simply a inverse now we are going to multiply both sides by D inverse and this will be post multiplication see in case of matrices if you have this or less than a equals to C then you can write a B equals to C B but you cannot write a B equals to BC this will be incorrect because on left hand side you are doing post multiplication and unlike on right hand side you are doing pre multiplication so either you can do post multiplication on both side or pre multiplication on both side so here I have multiplied both side by D inverse and this is post multiplication now D into D inverse this will be identity matrix so I will get B into C and this will be a inverse at D inverse now again since we have to separate B we are going to multiply by C inverse on both side and again this is post multiplication now B into C into c inverse this will be identity matrix so you have B equals to a inverse d inverse C inverse now the question was so this relation we have obtained now the question was we had to find the inverse of this matrix B so this means inverse of this matrix B this will be a inverse D inverse is C inverse inverse of this matrix now we know that in case of matrix multiplication we have to use the reversal law for inverse so now we have to write in reverse order C inverse inverse this will become C inverse inverse then d inverse inverse of this matrix then a inverse inverse now if you take inverse of a matrix twice then you will get the same matrix so this means inverse of matrix B this will be this will become C then inverse of D twice we will get two D then a so this is inverse of matrix B so in any type of these problem your first M will be you have to separate that that particular matrix and then you have to take inverse on both side so this was very basic problem now the next question is this question is based on the concept that we observe just learned now this question was asking Gaye 2016 so the problem is if M to the power 4 M is any matrix so M to the power 4 is an identity matrix and it is given that neither M nor M a square nor M cube is identity matrix then the question is for any natural number K we have to find M inverse so see it is given that M to the power 4 is identity matrix so I can write this relation as M into M cube this will be identity matrix now see the second matter we have two matrices one is M other is M cube and the product of these two matrices is identity matrix so this means the second matrix M cube is nothing but the inverse of first matrix so this is the equation that you have inverse of this matrix M this will be simply M cube now all the options are in the form of M to the power 4 k plus 1 for k plus 2 and so on so now we have to see which option is correct so if I place K equals to 0 here okay then you can see that you will get M to the power 3 here right I mean the inverse of this matrix is M inverse is M cube now if M inverse is M cube and M to the power 4 is identity so this means M to the power 7 this will be M to the power 4 into M cube M to the power 4 is identity matrix so this will be identity n to M Q versus matrix M cube similarly M to the power 11 this you can also write as m to the power 7 into M to the power 4 M to the power 7 we have already seen it is M cube M to the power 4 is identity so this will be M cube so if M inverse is M cube and all these values are same m to the power 7 is also M cube M to the power 11 is also M cube so inverse of this matrix will be M cube this is M to the power 7 which is M to the power 11 so all these matrices are inverse of matrix M so you can generalize this result and you can write that M inverse this will be nothing but you can see that this is 4 into 0 plus 3 this is 4 into 1 plus 7 sorry 4 into 1 plus 3 here you have 4 into 2 three so all these values you can write as 4 to the 4 k plus m and for a different value of K you will get all these numbers if you place K equals to 0 you will get M cube if you place K equals to 1 you will get M to the power 7 and so on so this will be the general formula for inverse of the matrix M inverse is M to the power 4 K plus 3 so the correct answer is option C now these two were very good question after this we are going to see some other columns based on inverse and adjoint of a matrix so these are the next two problem on inverse and adjoint of matrix this this question is a matrix a is given and 10 times matrix B is given and it is given that B is inverse of a then we have to find the value of alpha now see there are a number of ways to solve these type of problem one basic approach will be find the inverse of matrix a and then compare these two matrices it has given that B is inverse of a so simply by comparison you can find the value of alpha another method is that if a is if B is inverse of a so this means a into B this will be identity matrix so for my appropriate equation product of these two matrices a and B this will be identity so from here calculate the value of alpha so let me show you how to do this see if 10 B is this matrix this means B will be nothing but one tenth of this matrix right so you have one matrix a another matrix B and the product of these two matrices is identity matrix so I can write this relation as first our matrix a this will be 1 minus 1 1 2 1 minus 3 1 1 1 so this is our matrix a and matrix B is 1 by 10 this multiplied by 4 2 2 minus 5 0 alpha 1 minus 2 3 so you have one matrix a another matrix B and the product of these 2 matrices is identity matrix 1 0 0 0 1 0 then 0 0 1 so the product of these two matrices a and B this will be identity matrix now we have to find one that element which will give us an equation in alpha so maybe we can find the element in first row and third column so if I multiply the first row and the third column then I sort of get a zero right this multiplication of first row and third column this will give you the that particular element which is in Row one and column three so now if I multiply these numbers I will get one into 2 plus minus 1 into alpha plus 1 into 3 and you can see that element in first row third column is 0 so this is the equation that you have in alpha now you can easily solve this equation so 1 plus 2 or 3 sorry 2 into 1 2 plus 3 5 so from here you can see that the value of alpha that you will get is 5 so in this way you can easily solve you don't have to calculate the inverse of this matrix a simply write a into B this equals 2 identity matrix and then equate that particular element which will give you an equation in alpha for example you can also do 2nd row third column second row third column this would give you the element in second row third column this is 0 so from here you also you will get one linear equation in alpha and by solving that particular equation you can find the value of alpha so this was very simple question now the next one is this is based on the properties of adjoint and inverse that we have seen if a is any square matrix of order n such that determinant of adjoint of adjoint of a is determinant of a to the power 9 now the question is we have to find the value of n so we have seen that adjoint of adjoint of a this was nothing but determinant of a to the power n minus 2 into matrix a this was our formula now here the equation that you have is from this relation you can write determinant off and in place of adjoint of adjoint of a you can write determinant of a to the power n minus 2 into a so I will get determinant of a to the power n minus 2 into matrix a and this is nothing but determinant of a to the power 9 now see you have this matrix a a number determinant of a to the power n minus 2 so if I want to take this number outside then I have to take any power of this number outside we have discussed this property several times determinant of K into a this will be K to the power n into determinant of a so here if I want to take this number outside the sign of determinant I have to take any of this number outside and this multiplied by determinant of a this will be determinant of a to the power 9 so now you can see that this will be determinant of a to the power n minus 2 raised to the power n so this will be n into n minus 2 into determinant of here this will be determinant of a to the power 9 here I have used the property that property of indices a to the power M to the power n this is a to the power M into n so similarly we had n minus 2 raised to the power n so n into n minus 2 now we can write this as since the base is same we can add the exponent so this will be n into n minus 2 plus 1 this equals 2 determinant of a to the power 9 now we can equate the exponent so from here the equation that you will get is n into n minus 2 plus 1 this equals to 9 which means n square minus 2 n plus 1 minus 9 so this will be minus 8 equals to 0 which implies n minus 4 n plus 2 this equals to 0 which will give you n equals to minus 2 and plus 4 now obviously the order of a matrix cannot be negative so this minus 2 is not not acceptable and therefore the correct answer is 4 so this was very simple version once you know the properties of adjoint and inverse you can easily solve these type of problems so the correct answer for this question is 4 ok so now we are going to see these two problem on probability density function so the first question was asking to engineering services in 2017 the question is a random variable X has the probability density function FX equals to K into 1 by 1 plus X square X is from minus infinity to plus infinity the question is we have to find the value of K now we have learned that if here FX has given FX is k by 1 + X square and the range of X is from minus infinity to plus infinity now we have learned that if FX is probability density function then in that case integration minus infinity to infinity FX DX is 1 this was our third property if FX is probability density function then FX is always positive 2nd probability that X is from A to B is given by integral from A to B FX DX and this was the third point that integration from minus infinity to plus infinity FX DX this is always 1 so I can write this as integral minus infinity to infinity K by 1 + X square DX this will be 1 I can take K outside tan inverse X and the limit from minus infinity to plus infinity this will be 1 so this implies K into now if I place the upper limit I will get tan inverse infinity which is PI by 2 - if you place the lower limit then you will here tan inverse minus infinity which is minus PI by 2 so this value will be 1 so this means K into PI by 2 minus minus plus so this will become PI is 1 and finally you will get K equals to 1 by PI so the correct answer for this question is option B the value of K will be 1 by PI so these are very simple problem once you know that the the properties of probability density function or the definition of probability density function then you can easily solve these type of problems the second question is this one is also similar problem this was asked in 2016 in computer science the question is the probability density function on interval a to 1 is given by 1 by X square and outside this interval the value of function is 0 so this means our FX is this is the definition of FX FX is 1 by x square when x is from a to 1 on the interval a to 1 FX is 1 by X square so this is the value of FX when X is from a to 1 and is 0 otherwise so outside this interval the value of FX is 0 now the question is we have to find the value of this unknown a so again since FX's probability density function integral from minus infinity to infinity fxdx this value will be 1 so I can write this as integral from minus infinity to a FX DX plus integral from a to 1 FX DX plus integral from 1 to infinity FX DX this will be 1 I'm breaking this integral into 3 part minus infinity to a a to 1 1 to infinity now before a the value of FX is 0 ok FX has a nonzero well leave when X is from a to 1 so before a value of FX is 0 so this term will become 0 after 1 again the value of FX is 0 so this term will become 0 so basically we have this relation integral integral from a to 1 FX DX this value is 1 now when X is from a to 1 the value of FX is 1 by X square so I can write that integration a to 1 FX is 1 by X square DX this will be 1 now we have to solve this relation to find the value of a so integration of 1 by X this will be minus 1 by X and the limit is from a to 1 now in order to avoid calculus and mistake whenever you have this minus sign then convert this minus to plus and change the limit of integration so I will get the integration from 1 to a now this value is 1 and from here we have to find the value of a so you can see that this will give us 1 by a minus 1 is 1 this means 1 by a is 2 which means a is 1 by 2 which is 0.5 so therefore the correct answer for this question is a equals to 0.5 so these two were very simple problem once you know this relation that integration minus infinity to infinity FX DX is 1 if FX is probability density function then you can easily solve these type of problems our next concept is mean variance and standard deviation so first we will see how to calculate these quantities for discrete random variable and then we will see the same thing for continuous random variable so in case of discrete random variable the first step to calculate mean variance in a stand deviation will be first you have to draw probability distribution table so this will be the fastest step we have seen how to draw probability distribution table we have to draw - we have to create this table which will have two rows on first row we have to write the different value of x and in second row we have to write the corresponding probability so suppose the values of X are X 1 with probability P 1 X 2 with probability P 2 X 3 with probability P 3 and so on and the last value of x is x n and the corresponding probability is P n so first a step you have to make this table and now we are going to see how to calculate these quantities mean variance and a standard deviation so the first one is mean now this mean is also called expectation or expected value so mean or expectation is same thing and this is represented by either you can write X bar or you can write this as expectation of X so II X so the formula for this will be expectation of X will be simply P 1 X 1 you have to multiply the value of random variable ik X which is X 1 with the corresponding probability P 1 plus P 2 X 2 plus P 3 X 3 and so on up to P n X n so this will be the formula to calculate mean or expectation or expected value so expectation of X this you can also write in the form of summation summation P I X I so this will be the expected value or average value or mean of this random variable X now second thing is variance so second we have to calculate this variance now the variance is represented by Sigma s square and the formula to calculate variance will be this will be simply summation P I X I minus X bar s square summation P IX I minus X bar X bar is the mean so you have to subtract mean from each of the elements and then P ixi minus X bar as well this will give you the variance of the random variable X so for example if you want to write this in expanded form so this will be P 1 X 1 minus X bar square plus P 2 X 2 minus X bar square and so on up to P n xn minus X bar square so this will be the formula to calculate the variance so this X bar which is mean is constant that you have to take from here this value is nothing but X bar or mean so if you want to calculate the variance then you have to multiply P 1 with X 1 minus mean squared plus P 2 X 2 minus meter square and so on so this is one formula to find variance another formula for variance is another formula which is more popular is that if I have to calculate the variance then this will be summation P I and you can write this as X is square minus 2x I into X bar plus X bar square we are using the formula of a minus bi square so this will become now if I separate this then I will get some es NPI X is square minus X bar is constant so I can take this outside to time some SNP ixi plus again X bar square is constant summation P I so this will be the value of variance Sigma square now we know that sum of all the probabilities sum s and P I is nothing but to one we have seen this in our earlier discussion that if I add all the probability P 1 plus P 2 plus P 3 up to P n then I will get 1 and summation bi X is nothing but the mean so you will get summation P IX is square minus 2 X bar and this is also X bar means so X bar square plus this X bar square so finally you will get summation P IX is square minus X bar square so this is another formula to calculate mean mean Sigma square is P IX I square minus this mean square now sometimes this is also written as this variance Sigma square see some S&P ixi is expectation of X some S&P I X I this is expectation of X so here we have some S&P I X is square so this you can write as expectation of X square minus this X bar which is mean is nothing but expectation of X so I can write this as expectation of X s squared so this is again a very very important formula in for calculating variance of any random variable this Sigma square is expectation of X square minus expectation of X squared so so this was all about mean and variance and the last topic that and the last formula that we have to learn is a standard deviation so we can discuss that formula for a standard deviation here the third important point is a standard deviation a standard deviation so this standard deviation is represented by Sigma and this is nothing but a square root of variance so once you have calculated variance the standard deviation will be simply a square root of variance so these are the three formula that you have to remember the first one the first one was expectation of X or X bar and this was some s NPI X I so this was the first formula then second formula is for variance of X so this is the definition of variance but you generally we will be using the second formula to calculate variance and so and you can write the same formula in a slightly different manner as expectation of X square minus expectation of X square and finally the standard deviation is nothing but positive a square root of variance so these are the three formula now we will take one example and we will see how to calculate mean variance in a standard deviation so our first question is find mean variance and standard deviation of number of head find mean variance and standard deviation of number of hits in tossing two coins so if we toss two coins simultaneously and X is the random variable which is representing number of hits so the question is we have to find the mean variance in a standard deviation of number of hits so the first step will be we have to draw the probability distribution table now for that first we need to find the sample space and we have to see what are the different value of x and their corresponding probability X is representing number of head so if we are tossing two coins so we know that in that case the sample space will be head to head head tail tail head and tail tails so this will be the sample space these are the four possibility we are tossing two coins now X is representing number of head X is number of heads so the probability distribution table will be there are three possible value of x x is either 0 or 1 or 2 so first I am going to write X and then I will write the corresponding probability so the value of x will be either 0 if we are getting 2 tails for these two outcomes the value of x will be 1 and if we are getting 2 head then it means the value of x will be 2 so the so the possible value of x is either 0 or 1 or 2 and their corresponding probability will be x is 0 with probability 1 by 4 X is 1 with probability 2 by 4 and X is 2 with probability 1 by 4 so this will be the probability distribution table for number of head in tossing two coins now our job is we have to find mean variance and standard deviation of this X which is representing number of head so first one we are going to find the mean the formula for mean was this expectation of X in summation P I X I so this will mean P 1 X 1 plus P 2 X 2 plus P 3 X 3 so P 1 is 1 by 4 X 1 is 0 so this will be 1 by 4 into 0 plus P 2 is 2 by 4 X 2 is 1 so this will be 2 by 4 into 1 plus 1 by 4 into 2 so finally you will get a 0 plus 2 by 4 plus 2 by 4 this will be 4 by 4 which is 1 so this means you are tossing two coins then the average value value or expected value of number of head this will be 1 this simply means if you repeat the same experiment number of times then the average value of number of head that you are getting that will be 1 so this was the first result the second second one is we have to find the variance of number of heads so the formula for variance was this was summation P I X is square minus this mean square which is X bar so first thing this will be nothing this variance Sigma square this will be P 1 X 1 square plus P 2 X 2 s square plus P 3 X 3 square minus this mean s squared so now these are your P 1 P 2 P 3 X 1 X 2 X 3 so we have to substitute those values there so I will get 1 by 4 into 0 s square plus P 2 is 2 by 4 into 1 s square plus P 3 is 1 by 4 this multiplied by 2 s square minus mean is 1 so 1 squared so this will be 0 2 by 4 you will get 2 by 4 from here 4 by 4 this will be 1 minus 1 these two will cancel so finally your variance Sigma square is 1 by 2 so mean is 1 variance is 1 by 2 now we have to calculate the standard abs and also standard deviation Sigma this will be a square root of variance variance is 1 by 2 so this means the standard deviation Sigma this will be simply 1 by root 2 so if we are tossing two coins then in that case and if X is representing number of head then mean of X is 1 variance is 1 by 2 and standard deviation is 1 by root 1 by root 2 so after this we'll see how to calculate mean variance in the standard deviation for any continuous random variable okay so now we are going to see mean variance in a stand deviation for continuous random variable so suppose f FX is probability density function then in that case we have seen these three results the first one was FX is always positive if FX is probability density function then FX is positive for all x then second one was probability that X is from A to B this was given by the formula integration a to b fxdx and finally the value of integral from minus infinity to infinity FX DX this is 1 these things we already know but these are very important result so I'm writing those things again here now if FX is probability density function and we have to calculate the mean so the formula to calculate mean here will be mean or expectation this we will write as e X or X bar and this will meet this will mean the mean or the expected value so here the formula to calculate mean will be this will be simply integration minus infinity to infinity X FX DX using this formula you can find mean of any continuous random variable with a given probability density function so see this one is similar to this third result here your left hand side is integration minus infinity to infinity plus infinity fxdx now we are simply multiplying here by X so this will give you the mean or the expected value now one more important thing if you have to calculate expectation of X then you have X here so if I have to calculate the expectation of X s square so now this will be integration minus infinity to infinity and you have to replace X by X square so you will get X s square FX DX for X you have X here so for X square you will have X square here so this is the formula to calculate mean now the other two formula that we have to know is for variance and standard deviation so variance so the formula for variance this was represented by Sigma Square and this we have seen earlier this was expectation of X s square minus expectation of X squared this was the formula to calculate variants now we can easily once we know this formula we can easily find the variance using this concept expectation of X s square this will be integration minus infinity to infinity expectation of X is XFX DX so expectation of X s square this will be X square FX DX so this will be the first term minus the mean square mean you have already calculated using the first formula so we have to substitute that expectation or mean here so this will be the formula to find variance and the finally we have to see what should be the formula to calculate standard deviation so standard deviation we already know standard deviation Sigma has nothing but a square root of variance standard deviation this was represented by Sigma and this is positive square root of variance so first you have to get even if the problem is you have to find variance first you have to calculate the mean because in the formula of variance you have the mean here so first you have to calculate mean then you can find variance and finally your standard deviation will be a square root of variance so these are three very important formula that you have to know after this we'll take some problems and we'll see how to apply this formula