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 𝐴𝐴 and 𝐵𝐵 below, verify (check) the following
properties. Take 𝑛𝑛 = 3
I- Determinants:
i- 𝐴𝐴𝐴𝐴 = 𝐴𝐴 𝐵𝐵
ii- 𝐴𝐴𝑡𝑡 = 𝐴𝐴
iii- 𝑘𝑘𝐴𝐴3×3 = 𝑘𝑘 3 𝐴𝐴
iv- 𝐼𝐼 = 1
v- 0 = 0
vi- 𝐴𝐴 + 𝐵𝐵 ≠ 𝐴𝐴 + 𝐵𝐵
i- (𝐴𝐴𝐴𝐴)−1 = 𝐵𝐵−1 𝐴𝐴−1
1 −1
−1
ii-(𝑘𝑘𝑘𝑘) = 𝐴𝐴
𝑘𝑘
v-(𝐴𝐴 + 𝐵𝐵)−1 ≠ 𝐴𝐴−1 + 𝐵𝐵 −1
iii- (𝐴𝐴−1 )−1 = 𝐴𝐴
II- Inversions:
𝑡𝑡 )−1 = (𝐴𝐴−1 )𝑡𝑡
iv- (𝐴𝐴
vi-
𝐴𝐴−1
1
=
𝐴𝐴
, 𝐴𝐴 ≠ 0
For 𝑧𝑧1 and 𝑧𝑧2 below, verify (check) the following properties.
𝑧𝑧1 = −𝑖𝑖 + 3,
III- Complex Numbers:
𝑧𝑧2 = 4 − 2𝑖𝑖
𝑧𝑧1 +𝑧𝑧1
i- 𝑅𝑅𝑅𝑅(𝑧𝑧1 ) =
2
𝑧𝑧1 −𝑧𝑧1
ii- 𝐼𝐼𝐼𝐼(𝑧𝑧1 ) =
2𝑖𝑖
vii- 𝑧𝑧�1 = 𝑧𝑧1
viii- 𝑧𝑧�1 = 𝑧𝑧1
iv- 𝑧𝑧1 � 𝑧𝑧2 = 𝑧𝑧�1 � 𝑧𝑧�2
v-
𝑧𝑧1
𝑧𝑧2
𝑧𝑧1
= , (𝑧𝑧2 ≠ 0)
𝑧𝑧2
iii- 𝑧𝑧1 − 𝑧𝑧2 = 𝑧𝑧�1 − 𝑧𝑧�2
vi-
𝑧𝑧1
𝑧𝑧2
𝑧𝑧1
= , (𝑧𝑧2 ≠ 0)
𝑧𝑧2
ix- Sketch (Draw) 𝑧𝑧1 , 𝑧𝑧2 , 𝑧𝑧�1 and 𝑧𝑧�2 on the Cartesian or Argand plane.