Trace of the matrix [ 1^2 2^2 3^2 . . . . 20^2 2^2 3^2 .

Trace of the matrix [ 1^2 2^2 3^2 . . ....

Key Concepts

Trace of a Matrix

The trace of a matrix is defined as the sum of the elements along its main diagonal (from the top?left to the bottom?right). It is an important invariant in linear algebra and frequently appears in areas such as eigenvalue problems and characterizations of matrix properties.

Cyclic Structure in Matrices

When a matrix is constructed so that each row is a cyclic permutation of the first row, it possesses a cyclic structure. This symmetry means that while the rows are arranged in a shifted pattern, the manner in which the entries 'wrap around' can lead to predictable patterns—particularly in the diagonal entries—which can simplify computations such as finding the trace.

Arithmetic Sequences and Series Summation

Recognizing and summing the terms of an arithmetic sequence is a key technique in algebra. In problems where the diagonal entries form a sequence following a regular pattern (for example, a sequence of odd squares), knowing how to sum such series—whether by a formula or systematic addition—is essential to arriving at the correct result.

Transcript

00:01 So in the given question we are told to find the trace of a matrix that is given as the trace of the matrix which is given like this right so the matrix is having elements one square two square etc up to 20 square in the second row we have two square three square etc up to 1 square and this continues to the last row which is 20 square 1 square up to 19 square so first we should know what is a trace right so trace is nothing but the sum of the main diagonal elements in a matrix sum of main diagonal elements that is what we call a trace right so over here when we look at the elements in the main diagonal what we see is the elements one square three square and going on we would have five squared seven square etc up to 19 square right so what we see over here is that we are taking the sum of a certain number of consecutive odd numbers right so an expression for an odd number is 2 n minus 1 where n is any number from 1 to n right it can be 1 2 3 etc so when n equal to 1 we will have the first odd number which is 2 times 1 minus 1 which is equal to 1 and we should find what number in what question is 19 is right so 2n minus 1 is equal to 19 would give us 2n is equal to 20 and n equal to 10 which means 19 is the 10th odd number so what we are taking over here is that the trace of this matrix is the sum of the first 10 odd numbers right so we can write this expression as sigma of n equal to 1 to 10 2 n minus 1 whole squared right and this would be equal to sigma n equal to 1 to 10 4 n squared we are expanding this so 4 n squared minus sigma n equal to 1 to 10 4 n plus sigma n equal to 1 to 10 1 so we should know the expansion for the first n natural numbers the sum of first n natural numbers sum of first n natural numbers is given as n times n plus 1 times 2 n plus 1 divided by 6 and to find the sum of this is the sum of squares of squares of the of the first n natural numbers, right? you should know that this is the sum of, sum of squares of, sum of squares of first n natural numbers is given by this formula and the sum of first n natural numbers that is without taking the square if you are adding the first n natural numbers what we would have is n into n plus 1 by 2 so over here we can substitute for sigma of n square as we can take 4 as a factor and write the expression as 4 times sigma of n equal to 1 to 10 n square minus 4 times sigma n equal to 1 to 10 of n plus sigma n equal to 1 to 10 of 1.

04:37 So now using the formula we can substitute this expression over here by substituting for n in the above formula as 10.

04:49 Right so for substitute for taking the sum of first 10 natural numbers that is from 1 to 10 the formula would be 10 times 10 plus 1 which is 11 times 2 n plus 1 which is 2 times 10 which is 20 and 20 plus 1 21 divided by 6 minus we have 4 times the first sum of first 10 10 numbers which can be taken us 10 times 10 plus 1 times which is 11 divided by 2 and then we have the sum of first we have the sum of 1 that is from n equal to 1 to 10 we have to add 1 10 times right which would give us 10 so what we would have over here is when we calculate 4 times 10 times 11 times 21 divided by 6 we will get 1 ,000 by 140 minus we have 4 times 10 times 11 divided by 2 which is 220 and plus 10 and this would give us the required answer or the trace of this matrix as 1 ,330...

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