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a) Using augmented form
Step 1: Multiply the first row by -3/10 and substract from the second line, Multiply the first row by 1/10 and substract from the third line,
Step 2: Multiply the second row by 0.8/-5.4 and substract from the third line,
x3 = (-30.278/5.352) = -5. x2 = (-52.5-1.7(-5.657))/(-5.4) = 7. x1 = (25-27.941-(-1)*(-5.657))/10 = 0.
b) [A] =
൩ b = ൭
Construct the upper and lower triangular matrices. [U] =
൩ [L] =
[L] {d}={b} and [U] {x}={d}
d1 = 25, d2 = -52.5, d=-30. {d} = ൭
[U] =
x3 = -5.656, x2 = 7.942, x1 = 0. {x} = ൭
c) [L] [U] ቌ
(𝑎ି ଵ^ )ଵଵ
(𝑎ି ଵ^ )ଶଵ
(𝑎ି ଵ^ )ଷଵ
൱ [L]{d}={I 1 } = ൭
[L] [U] ቌ
(𝑎ି ଵ^ )ଵଶ
(𝑎ି ଵ^ )ଶଶ
(𝑎ି ଵ^ )ଷଶ
൱ [L]{d}={I 2 } = ൭
[L] [U] ቌ
(𝑎ି ଵ^ )ଵଷ
(𝑎ି ଵ^ )ଶଷ
(𝑎ି ଵ^ )ଷଷ
൱ [L]{d}={I 3 } = ൭
[U]{x}={d} ቌ
(𝑎ି ଵ^ )ଵଵ
(𝑎ି ଵ^ )ଶଵ
(𝑎ି ଵ^ )ଷଵ
[U]{x}={d} ቌ
(𝑎ି ଵ^ )ଵଶ
(𝑎ି ଵ^ )ଶଶ
(𝑎ି ଵ^ )ଷଶ
[U]{x}={d} ቌ
(𝑎ି ଵ^ )ଵଷ
(𝑎ି ଵ^ )ଶଷ
(𝑎ି ଵ^ )ଷଷ
[A]-1^ =
a) Using augmented form
Step 1: Multiply the first row by -3/10 and substract from the second line, Multiply the first row by 1/10 and substract from the third line,
Step 2: Multiply the second row by 0.8/-5.4 and substract from the third line,
x3 = (-43.75/26.666) = -1. x2 = (0-12.5(-1.640666))/(30) = 0. x1 = (50-15(-1.640666)-20*(0.6836108))/8 = 7. b) [A] =
൩ b = ൭
൱ [U] =
൩ [L] =
[L] {d}={b} and [U] {x}={d}
d1 = 50, d2 = 0, d=-43.75; {d} = ൭
[U] =
x3 = -1.6406, x2= 0.6836, x1=7. {x} = ൭
