math notes for middle school

In the above method note that To obtain D 1 , replace a 1 , a 2 , a 3 by d 1 , d 2 , d 3 in D To obtain D 2 , replace b 1 , b 2 , b 3 by d 1 , d 2 , d 3 in D To obtain D 3 , replace c 1 , c 2 , c 3 by d 1 , d 2 , d 3 in..

(i) Note that the entries in a given matrix need not be distinct. (ii) The entries in this matrix are function of x. A matrix having m rows and n columns is called as matrix of order mxn. Such a matrix has mn elements. In general, an mxn matrix is in the form Where a i j represen..

>If A, B and C are the matrices which can be multiplied then (a) Matrix multiplication is not commutative, i.e., AB BA (always) (b) Associative law holds good for matrix multiplication, i.e., (AB)C = A(BC) (c) Matrix multiplication is distributive with respect to addition A(B + C) = AB + AC or (A +..

Note that the position is 1 s t row and 2 n d column. Delete the 1 s t row and 2 n d column, the determinant so obtained is the minor of a 1 2 That is minor of a 1 2 The co-factor of a 1 2 = A 1 2 = (-1) 1 + 2 M 1 2 = - (a 2 1 a 3 3 - a 3 1 a 2 3 ) Similarly the minor of a 1 3 = a 2 1 a 3..

Strictly speaking 0! has no meaning. But since n P n = n! we may understand 0! = ..

i) The series formed by the reciprocals of the terms of a geometric series is also a geometric series. ii) There is no general method of finding the sum of a harmonic progressi..

From the definition, it is clear that if B is the inverse of A, then A is the inverse of ..

If the product of three numbers in GP is given, take the term as a/r, a, ar. But if the product of the numbers is not given, the terms are in the ordinary for..

Question: Answer: ..