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Symmetric Matrix
A square matrix A = [a i j ] is said to be symmetric if its (i,j) t h element is the same as its (j,i) t h element. i.e a i j = a j i " i, j A square matrix A is said to be symmetric, if A = A ..
A square matrix A = [a i j ] is said to be symmetric if its (i,j) t h element is the same as its (j,i) t h element. i.e a i j = a j i " i, j A square matrix A is said to be symmetric, if A = A ..Examples:
skew-symmetric for every square matrix A. That is any square matrix is expressible as the sum of a symmetric matrix and a skew-symmetric matrix. 5. A matrix which is both symmetric and skew symmetric is a zero matr..
skew-symmetric for every square matrix A. That is any square matrix is expressible as the sum of a symmetric matrix and a skew-symmetric matrix. 5. A matrix which is both symmetric and skew symmetric is a zero matr..
..Diagonal Matrix
A square matrix A=[a i j ] n x n is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero. i.e., a i j = 0 for all i ..
A square matrix A=[a i j ] n x n is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero. i.e., a i j = 0 for all i ..Example:
The matrices are identify matrices of order 2 and 3 respectivel..
The matrices are identify matrices of order 2 and 3 respectivel..Proof:
Each circular permutation corresponds to n linear permutations depending on where we start. Since there are exactly n! linear permutations, there are exactly permutations. Hence, the number of circular permutations is the same as (n-1)..
Each circular permutation corresponds to n linear permutations depending on where we start. Since there are exactly n! linear permutations, there are exactly permutations. Hence, the number of circular permutations is the same as (n-1)..Theorem:
The number of combinations of n dissimilar things, taken r at a time is..
The number of combinations of n dissimilar things, taken r at a time is..Proof:
If r = s, there is nothing to prove. Now, If r < s, then n - r > n - s, then the above equation becomes Since both sides are products of (s-r), consecutive integers in Similarly it can be proved that n = r + s if r > s...
If r = s, there is nothing to prove. Now, If r < s, then n - r > n - s, then the above equation becomes Since both sides are products of (s-r), consecutive integers in Similarly it can be proved that n = r + s if r > s...Proof:
..
..Examples:
8! = 8 x 7 x 6 x 5 x4 x 3 x 2 x 1 = 40320 ..
8! = 8 x 7 x 6 x 5 x4 x 3 x 2 x 1 = 40320 ..See what our Users say :
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