Properties of Integral 3

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BASICS OF MATRIX

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  Addition, scalar multiplication, subtraction and transposition [ edit ] Addition The  sum   A  +  B  of two  m × n  matrices  A  and  B  is calculated entrywise: ( A + B ) i , j = A i , j + B i , j , 1 ≤ i ≤ m , 1 ≤ j ≤ n . For example, [ 1 3 1 1 0 0 ] + [ 0 0 5 7 5 0 ] = [ 1 + 0 3 + 0 1 + 5 1 + 7 0 + 5 0 + 0 ] = [ 1 3 6 8 5 0 ] Scalar multiplication The product  c A  of a number  c  (also called a  scalar  in this context) and a matrix  A  is computed by multiplying every entry of  A  by  c : ( c A ) i , j = c ⋅ A i , j This operation is called  scalar multiplication , but its result is not named "scalar product" to avoid confusion, since "scalar product" is often used as a synonym for " inner product ". For example: 2 ⋅ [ 1 8 − 3 4 − 2 5 ] = [ 2 ⋅ 1 2 ⋅ 8 2 ⋅ − 3 2 ⋅ 4 2 ⋅ − 2 2 ⋅ 5 ] = [ 2 16 − 6 8 − 4 10 ] Subtraction The subtraction of two  m × n  matr...
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ALGEBRAIC IDENTITIES

  Standard Algebraic Identities List All standard Algebraic Identities are derived from the  Binomial Theorem . There are a number of algebraic identities but few are standard that are listed below. (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 + b 2 – 2ab (a + b)(a – b) = a 2 – b 2 (a + b) 3 =a 3 + b 3 + 3ab(a + b) (a – b) 3 =a 3 – b 3 – 3ab(a – b) (a + b + c) 2 =a 2 + b 2 + c 2 + 2ab + 2bc + 2ca Methods to Solve  Algebraic Identities We can verify algebraic identities by  substitution method , in which we can put values in variable places and try to make both sides equal. i.e LHS = RHS. Example: (a – 2) (a + 2) = a 2 – 2 2 Now we will start putting value in place of a. starting with a = 1, (-1) x (3) = -3 then we will put a = 2, 0 x 4 = 0 Here we got a = 1 and a = 2 as the value which satisfy the given question. Another method is by  manipulating identities  which are commonly used: i. (a + b) 2 = a 2 + b 2 + 2ab     ...
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