Higher Engineering Mathematics

262 COMPLEX NUMBERS Determine in polar and cartesian forms (a) [3∠ 41 ◦]^4 (b) (− 2 −j)^5. [ (a) 81∠ 164 ◦,− 77. 86 +j 22. 33 ( ...

DE MOIVRE’S THEOREM 263 E but the arguments,θ, are different. It is shown in Problem 3 that arguments are symmetrically spaced o ...

264 COMPLEX NUMBERS (− 2 +j) 1 4 ⎡ ⎣ Moduli 1.223, arguments 38 ◦ 22 ′, 128◦ 22 ′, 218 ◦ 22 ′and 308◦ 22 ′ ⎤ ⎦ (− 6 −j5) 1 ( ...

DE MOIVRE’S THEOREM 265 E Problem 6. Change (3−j4) into (a) polar form, (b) exponential form. (a) (3−j4)= 5 ∠− 53. 13 ◦or 5 ∠− 0 ...

266 COMPLEX NUMBERS Express 2e^3 +j π (^6) in (a+jb) form. [34. 79 +j 20 .09] Convert 1.7e^1.^2 −j^2.^5 into rectangular form ...

Matrices and Determinants F 25 The theory of matrices and determinants 25.1 Matrix notation Matrices and determinants are mainly ...

268 MATRICES AND DETERMINANTS Problem 2. Subtract (a) ( − 30 7 − 4 ) from ( 2 − 1 − 74 ) and (b) ( 27 − 5 −21 0 63 4 ) from ( 31 ...

THE THEORY OF MATRICES AND DETERMINANTS 269 F Problem 5. IfA= ( 23 1 − 4 ) andB= ( − 57 − 34 ) findA×B. LetA×B=CwhereC= ( C 11 C ...

270 MATRICES AND DETERMINANTS In algebra, the commutative law of multiplication states thata×b=b×a. For matrices, this law is on ...

THE THEORY OF MATRICES AND DETERMINANTS 271 F 11.E×K ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 3 1 2 6 12 − 2 3 − 2 5 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 12 ...

272 MATRICES AND DETERMINANTS 25.5 The inverse or reciprocal of a 2 by 2 matrix The inverse of matrixAisA−^1 such thatA×A−^1 =I, ...

THE THEORY OF MATRICES AND DETERMINANTS 273 F Determine the inverse of ( − 1. 37. 4 2. 5 − 3. 9 ) ⎡ ⎣ ( 0 .290 0. 551 0 .186 0 ...

274 MATRICES AND DETERMINANTS Problem 16. Determine the value of ∣ ∣ ∣ ∣ ∣ j2(1+j)3 (1−j)1j 0 j 45 ∣ ∣ ∣ ∣ ∣ Using the first col ...

THE THEORY OF MATRICES AND DETERMINANTS 275 F Problem 17. Determine the inverse of the matrix ⎛ ⎝ 34 − 1 207 1 − 3 − 2 ⎞ ⎠ The i ...

276 MATRICES AND DETERMINANTS Determine the adjoint of ( 4 − 76 − 240 57 − 4 ) [( − 16 14 − 24 − 8 − 46 − 12 − 34 − 63 2 )] D ...

F Matrices and Determinants 26 The solution of simultaneous equations by matrices and determinants 26.1 Solution of simultaneous ...

278 MATRICES AND DETERMINANTS (b) The procedure for solving linear simulta- neous equations inthree unknowns using matricesis: ( ...

THE SOLUTION OF SIMULTANEOUS EQUATIONS BY MATRICES AND DETERMINANTS 279 F Now try the following exercise. Exercise 113 Further p ...

280 MATRICES AND DETERMINANTS Following the above procedure: (i) 3x− 4 y− 12 = 0 7 x+ 5 y− 6. 5 = 0 (ii) x ∣ ∣ ∣ ∣ − 4 − 12 5 − ...

THE SOLUTION OF SIMULTANEOUS EQUATIONS BY MATRICES AND DETERMINANTS 281 F I 1 (− 20 +j40)+(40+j15) = −I 2 (30−j60)−(30+j40) = 1 ...