Guass Jordan

[oblique umbra]

I understand that it's r1-r3 but how can i display that as a elementary matrix i keep getting it wrong

[oblique umbra]

here are the two sections of it

[thorn isle]

the row operations are a shorthand of using the elementary matrix.

you know that any matrix multiplied to the identity matrix will result to the same matrix.

and from matrix multiplication, you'll iterate from each entry in the identity matrix's rows to your matrix's cols to get one entry.

[thorn isle]

say your first step

[
\begin{bmatrix}
1 & 0 & 0 \
0 & \frac{1}{8} & 0 \
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
2 & 0 & 1 \
80 & 8 & 40 \
1 & 0 & 0
\end{bmatrix}
]

[jaunty badger]

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[thorn isle]

So for your first row it'll look something like
$
R1 = r1 \
I_{1 ; 1} = 1(2) + 0(80) + 0(1) = 2 \
I_{1 ; 2} = 1(0) + 0(8) + 0(0) = 0 \
I_{1 ; 3} = 1(1) + 0(40) + 0(0) = 1 \
$

\begin{bmatrix}
2 & 0 & 1
\end{bmatrix}

[jaunty badger]

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[thorn isle]

But once you reach the row where you changed the coefficient (e.g. 1/8)
that row gets affect
$
R_{2} = (1/8)r_{2} \
I_{2 ; 1} = 0(2) + (1/8)(80) + 0(1) = 10 \
I_{2 ; 2} = 0(0) + (1/8)(8) + 0(0) = 1 \
I_{2 ; 3} = 0(1) + (1/8)(40) + 0(0) = 8
$

$
\begin{bmatrix}
10 & 1 & 8
\end{bmatrix}
$

[jaunty badger]

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[thorn isle]

now suppose you want to swap the places of the two rows
well, to reflect this change you have to swap the rows of your elementary matrix as well

R3 -> r1
R1 -> r3

$
E = \begin{bmatrix}
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0
\end{bmatrix}
$

[jaunty badger]

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[thorn isle]

to add a row to another row, you can do the elementary matrix like this

[
R_1 \to R_1 - 2R_3
]

Given:

[
EA =
\begin{bmatrix}
1 & 0 & -2 \
0 & 1 & 0 \
0 & 0 & 1 \
\end{bmatrix}
\begin{bmatrix}
2 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
\end{bmatrix}
]

[jaunty badger]

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[thorn isle]

and it goes for your first row as something like

I_{1 ; 1} = 1(2) + 0 + -2(1) = 0
I_{1 ; 2} = 1(0) + 0(1) + -2(0) = 0
I_{1 ; 3} = 1(1) + 0 + -2(0) = 1

and get [ 0 , 0, 1 ]