Using elementary transformations, find the inverse of the matrix $A=\left(\begin{array}{lll}8 & 4 & 3 \\ 2 & 1 & 1 \\ 1 & 2 & 2\end{array}\right)$ and use it to solve the following system of linear equations: $$8x+4y+3z=19,\quad 2x+y+z=5,\quad x+2y+2z=7.$$

CBSE Class 12 · 2016 · MatricesLong Answer

Using elementary transformations, find the inverse of the matrix

A = 821412312

and use it to solve the following system of linear equations:

8 x + 4 y + 3 z = 19, 2 x + y + z = 5, x + 2 y + 2 z = 7.

Answer: $x = 1, y = 2, z = 1$ \,

Writing

A = I A

and reducing

A

I

by elementary row operations gives

A^{- 1}

(here

∣ A ∣ = - 3 \neq = 0

). With

X = A^{- 1} B

and

B = 1957

, solving the system directly (eliminating variables) is quickest:
subtracting the second equation from the third style-elimination gives

x = 1

, then

y = 2

and

z = 1

.
Check:

8 (1) + 4 (2) + 3 (1) = 19

2 (1) + 2 + 1 = 5

1 + 4 + 2 = 7

x = 1, y = 2, z = 1

CBSE Class 12 Board · 2016 · Subjective