Givem-that-the-matrix-A-3-1-5-2-3-5-5-1-6-If-Adj-A-13-1-10-13-7-5-13-2-7-i-find-A-1-ii-Use-the-result-in-i-to-find-the-values-of-x-y-and-z-that

Question Number 205586 by necx122 last updated on 25/Mar/24

G i v e m t h a t t h e m a t r i x A = (\begin{matrix} 3 & 1 & 5 \\ 2 & 3 & 5 \\ 5 & 1 & 6 \end{matrix}) .

I f A d j . A = (\begin{matrix} 13 & - 1 & - 10 \\ 13 & - 7 & - 5 \\ - 13 & 2 & 7 \end{matrix})

(i) f i n d A^{- 1}

(i i) U s e t h e r e s u l t i n (i) t o f i n d t h e

v a l u e s o f x, y a n d z t h a t w i l l s a t i s f y t h e

e q u a t i o n s :

3 x + y + 5 z = 8

2 x + 3 y + 5 z = 0

5 x + y + 6 z = 13

Answered by som(math1967) last updated on 25/Mar/24

A^{- 1} = \frac{1}{∣ A ∣} \times A d j . A

∣ A ∣= 3 (18 - 5) - 1 (12 - 25) + 5 (2 - 15)

= 39 + 13 - 65

= 13 (3 + 1 - 5) = - 13

∴ A^{- 1} = (\begin{matrix} - 1 & \frac{1}{13} & \frac{10}{13} \\ - 1 & \frac{7}{13} & \frac{5}{13} \\ 1 & \frac{- 2}{13} & \frac{- 7}{13} \end{matrix})

e q u a t i o n

A \times X = B

w h e r e = A = (\begin{matrix} 3 & 1 & 5 \\ 2 & 3 & 5 \\ 5 & 1 & 6 \end{matrix})

X = (\begin{matrix} x \\ y \\ z \end{matrix}) B = (\begin{matrix} 8 \\ 0 \\ 13 \end{matrix})

A^{- 1} \times A \times X = A^{- 1} \times B

I \times X = (\begin{matrix} - 8 + 0 + 10 \\ - 8 + 0 + 5 \\ 8 + 0 - 7 \end{matrix})

(\begin{matrix} x \\ y \\ z \end{matrix}) = (\begin{matrix} 2 \\ - 3 \\ 1 \end{matrix})

∴ x = 2, y = - 3, z = 1 a n s

Commented by necx122 last updated on 25/Mar/24

T h i s i s s o c l e a r a n d e x p r e s s i v e . T h a n k y o u

s o m u c h s i r .

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