If X = [ x y z ] is a solution of the system of equations AX=B, where A = [ 4 2 2 - 5 0 5 1 - 2 3 ] a n d B = [ 4 0 2 ] then | x + y + z | is equal to: [2026]

Question

If $X = [[\begin{matrix} x \\ y \\ z \end{matrix}]]$ is a solution of the system of equations AX=B, where adj $A = [[\begin{matrix} 4 & 2 & 2 \\ - 5 & 0 & 5 \\ 1 & - 2 & 3 \end{matrix}]] and B = [[\begin{matrix} 4 \\ 0 \\ 2 \end{matrix}]]$ then $| x + y + z |$ is equal to: [2026]

Smart Achievers · Accepted Answer

(2)

$X = A^{- 1} B = (\frac{adj A}{| A |}) B$

$= \pm \frac{1}{10} (\begin{matrix} 4 & 2 & 2 \\ - 5 & 0 & 5 \\ 1 & - 2 & 3 \end{matrix}) (\begin{matrix} 4 \\ 0 \\ 2 \end{matrix})$

$= \pm \frac{1}{10} (\begin{matrix} 20 \\ - 10 \\ 10 \end{matrix}) = \pm (\begin{matrix} 2 \\ - 1 \\ 1 \end{matrix})$

$∴$ $| x + y + z |$ $= 2$

Solution

Q.
If $X = [[\begin{matrix} x \\ y \\ z \end{matrix}]]$ is a solution of the system of equations AX=B, where adj $A = [[\begin{matrix} 4 & 2 & 2 \\ - 5 & 0 & 5 \\ 1 & - 2 & 3 \end{matrix}]] and B = [[\begin{matrix} 4 \\ 0 \\ 2 \end{matrix}]]$ then $| x + y + z |$ is equal to: [2026]

1 3/2

2 2

3 1

4 3

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Solution

Q. If X=[xyz] is a solution of the system of equations AX=B, where adj A=[422-5051-23] and B=[402] then |x+y+z| is equal to: [2026]

1 3/2

2 2

3 1

4 3

Ans. (2) X=A-1B=(adjA|A|)B =±110(422-5051-23)(402) =±110(20-1010)=±(2-11) ∴ |x+y+z|=2

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Q.
If $X = [[\begin{matrix} x \\ y \\ z \end{matrix}]]$ is a solution of the system of equations AX=B, where adj $A = [[\begin{matrix} 4 & 2 & 2 \\ - 5 & 0 & 5 \\ 1 & - 2 & 3 \end{matrix}]] and B = [[\begin{matrix} 4 \\ 0 \\ 2 \end{matrix}]]$ then $| x + y + z |$ is equal to: [2026]