Understanding Engineering Mathematics

12.If|z 1 |=5, Argz 1 =π/3,|z 2 |=3, Argz 2 =π/4, find the Cartesian forms ofz 1 and z 2 and the values of: (i) |z 1 z 2 | (ii) ...

19.Simplify (i)( 1 +j √ 3 )^6 +( 1 −j √ 3 )^6 and (ii) ( √ 3 −j)^15 by using De Moivre’s theorem. 20.Simplify (cos 3θ−jsin 3θ)−^ ...

3.In the relationship (R+jpL) ( S−j 1 pC ) = P Q all the quantities are real exceptj. Show that p= √ R LSC and findRin terms ofC ...

In terms ofXwe can writex(t)as: x(t)=Im(Xejωt) wherein lies the usefulness of phasors – the frequency behaviour of x(t) is sepa- ...

12.11 Answers to reinforcement exercises (i) j (ii) 3 (iii) 3j (iv) −1(v)1(vi)−j (vii) − 1 (i) ± 5 j (ii) − 2 +j,− 2 −j (iii) ...

− 2 + 2 j 3 j − j 1 + j√ 3 −√ 3 − j − 3 − 3 j − (^102) y x 11.(i) 4 (ii) − 3 j (iii) − 2 (iv) − 10 (v) 10j (vi) √ 2 + √ 2 j (vii ...

2 j − 3 j 1 −j (^0) x y √ 3 2 1 2 j 1 2 +j √ 3 2 19.(i) 2^7 (ii) − 215 j cos( 38 θ)−jsin( 38 θ) (i) ± 1 (ii) ±j (iii) ...

23.(i) cos 7α+jsin 7α (ii) sin 2α−jcos 2α (iii) sinα+jcosα 2 ( π+ 2 kπ 4 ) or± √ 2 ( 1 ±j) √2 (− 1 + j) √2 (1+ j) √2 (1− j) √ ...

13 Matrices and Determinants Matrices are rectangular arrays of numbers treated as mathematical entities in themselves and satis ...

13.1 An overview of matrices and determinants Everywhere in engineering and science we are constantly having to deal with situat ...

must be properly included. The elements can be complex numbers. What we want to do is construct an ‘algebra’ for such arrays tha ...

(i) By analogy with how we have rewritten the pairs of equations we ‘multiply’ by plugging rows of the first matrix into columns ...

Multiplication of matrices, considered in Section 13.3, isassociative: A(BC)=(AB)C butnot commutative: AB=BA We also define som ...

are the same, i.e. [bij]=[aij] if and only ifbij=aij for every value ofiandj. Weadd/subtractmatrices by adding/subtracting corre ...

Problem 13.4 If [ ab cd ] Y [ 3 − 1 10 ] = [ 10 01 ] finda,b,c,d. We have [ ab cd ] + [ 3 − 1 10 ] = [ a+ 3 b− 1 c+ 1 d+ 0 ] = [ ...

That is, the element in theith row andjth column of the product matrixCis obtained from the ‘scalar’ product of theith row ofAwi ...

[− 123 011 − 102 ][ 2 − 4 − 1 − 111 1 − 2 − 1 ] = [− 100 0 − 10 00 − 1 ] =− [ 100 010 001 ] =−I whereIis called the 3× 3 unit ma ...

For the matrices A= [−10 1 21 0 32 − 1 ] B= [ 423 − 241 321 ] evaluate (i) A+B (ii) A−B (iii) 3A+ 2 B (iv) AB (v) BA (vi) A^2 ...

So x 2 = a 11 b 2 −a 21 b 1 a 11 a 22 −a 21 a 12 Similarly we find x 1 = b 1 a 22 −a 12 b 2 a 11 a 22 −a 21 a 12 Introduce the n ...

order determinants to fit in with the above pattern. The simplest way to define these higher order determinants is in terms of s ...