Solve the following systems Using (I) Row reduced Echelon Form
(II) Jauss-Jordan Elimination and (III) by determining the inverse of the
matrix of coefficients and then using matrix multiplication.
For the matrices π΄π΄ and π΅π΅ below, verify (check) the following
properties. Take ππ = 3
I- Determinants:
i- π΄π΄π΄π΄ = π΄π΄ π΅π΅
ii- π΄π΄π‘π‘ = π΄π΄
iii- πππ΄π΄3×3 = ππ 3 π΄π΄
iv- πΌπΌ = 1
v- 0 = 0
vi- π΄π΄ + π΅π΅ ≠ π΄π΄ + π΅π΅
i- (π΄π΄π΄π΄)−1 = π΅π΅−1 π΄π΄−1
1 −1
−1
ii-(ππππ) = π΄π΄
ππ
v-(π΄π΄ + π΅π΅)−1 ≠ π΄π΄−1 + π΅π΅ −1
iii- (π΄π΄−1 )−1 = π΄π΄
II- Inversions:
π‘π‘ )−1 = (π΄π΄−1 )π‘π‘
iv- (π΄π΄
vi-
π΄π΄−1
1
=
π΄π΄
, π΄π΄ ≠ 0
For π§π§1 and π§π§2 below, verify (check) the following properties.
π§π§1 = −ππ + 3,
III- Complex Numbers:
π§π§2 = 4 − 2ππ
π§π§1 +π§π§1
i- π
π
π
π
(π§π§1 ) =
2
π§π§1 −π§π§1
ii- πΌπΌπΌπΌ(π§π§1 ) =
2ππ
vii- π§π§οΏ½1 = π§π§1
viii- π§π§οΏ½1 = π§π§1
iv- π§π§1 οΏ½ π§π§2 = π§π§οΏ½1 οΏ½ π§π§οΏ½2
v-
π§π§1
π§π§2
π§π§1
= , (π§π§2 ≠ 0)
π§π§2
iii- π§π§1 − π§π§2 = π§π§οΏ½1 − π§π§οΏ½2
vi-
π§π§1
π§π§2
π§π§1
= , (π§π§2 ≠ 0)
π§π§2
ix- Sketch (Draw) π§π§1 , π§π§2 , π§π§οΏ½1 and π§π§οΏ½2 on the Cartesian or Argand plane.
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