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Elementary Transformations

A =[[ 1 5 2; ...

Question

$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 5 & 2 3 & 1 & 4 2 & 2 & 3 \end{matrix} ⎤ ⎥ ⎦$
The following elementary transformations are applied on the matrix A in the given order. The resultant matrix after the following operations $R_{2}$ $\to$ $R_{2}$ – $R_{3}$ , $R_{1}$ $\to$ $R_{1} - R_{2}$ , $R_{3}$ $\to$ $R_{3}$ – 2 $R_{2}$ is

⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 1 2 & - 1 & 2 0 & - 4 & 1 \end{matrix} ⎤ ⎥ ⎦

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⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 1 1 & - 1 & 1 0 & - 4 & 1 \end{matrix} ⎤ ⎥ ⎦

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⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 2 2 & 1 & 1 0 & - 4 & 1 \end{matrix} ⎤ ⎥ ⎦

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⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 2 1 & 1 & 1 0 & - 4 & 1 \end{matrix} ⎤ ⎥ ⎦

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Solution

The correct option is B $⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 1 1 & - 1 & 1 0 & - 4 & 1 \end{matrix} ⎤ ⎥ ⎦$
Given matrix, $⎡ ⎢ ⎣ \begin{matrix} 1 & 5 & 2 3 & 1 & 4 2 & 2 & 3 \end{matrix} ⎤ ⎥ ⎦$

$R_{2} \to R_{2} - R_{3}$
$⎡ ⎢ ⎣ \begin{matrix} 1 & 5 & 2 3 - 2 & 1 - 2 & 4 - 3 2 & 2 & 3 \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎣ \begin{matrix} 1 & 5 & 2 1 & - 1 & 1 2 & 2 & 3 \end{matrix} ⎤ ⎥ ⎦$

$R_{1} \to R_{1} - R_{2}$
$⎡ ⎢ ⎣ \begin{matrix} 1 - 1 & 5 + 1 & 2 - 1 1 & - 1 & 1 2 & 2 & 3 \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 1 1 & - 1 & 1 2 & 2 & 3 \end{matrix} ⎤ ⎥ ⎦$

$R_{3} \to R_{3} - 2 R_{2}$
$⎡ ⎢ ⎣ \begin{matrix} 0 & 6 & 1 1 & - 1 & 1 0 & 4 & 1 \end{matrix} ⎤ ⎥ ⎦$

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Similar questions

Applying the following transformations will result in on the matrix in order on the matrix in order $⎡ ⎢ ⎣ \begin{matrix} 1 & 5 & 2 3 & 1 & 4 2 & 2 & 3 \end{matrix} ⎤ ⎥ ⎦$

Q. Assertion :In a

△ A B C

, if

a < b < c

and

r

is inradius and

r_{1}, r_{2}, r_{3}

are the axradii opposite to angle

A, B, C

respectively, then

r < r_{1} < r_{2} < r_{3}

Reason:

△ A B C, r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = \frac{r_{1} r_{2} r_{3}}{r}

Q. Assertion :In a

Δ

ABC,

r_{1} + r_{2} + r_{3} - r = 4 R

Reason:

r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = Δ^{2}

\frac{R_{1}}{r_{1}} + \frac{R_{2}}{r_{2}} + \frac{R_{3}}{r_{3}} =

Q. If in a

△ A B C, r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}

is equal to
(where

r_{1}, r_{2}

and

r_{3}

are the exradii and

2 s

is the perimeter.)

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A=⎡⎢⎣152314223⎤⎥⎦ The following elementary transformations are applied on the matrix A in the given order. The resultant matrix after the following operations R2 → R2 – R3, R1 → R1–R2, R3 → R3 – 2R2 is