The matrix is obtained from the unit matrix I by the elementary row operation of

Question

The matrix is obtained from the unit matrix I by the elementary row operation of adding c times its first row to its second row. Show that for every 3 times 3 matrix A the same elementary row operation performed on A results in the product matrix EA. Also, find E-1 and describe the elementary row operation it corresponds to. (A matrix that produces the same effect by multiplication as an elementary row operation, like this E and the matrices P in the next two exercises, is called an elementary matrix.)

Explanation / Answer

E =
1 0 0
c 1 0
0 0 1

now

let take A matrix

A =

a11 a12 a13
a21 a22 a23
a31 a32 a33

now given elementary operation is add c times first row to second row.

R2 = R2 + C R1

a11 a12 a13
ca11+ a21 ca12 +a22 ca13 + a23
a31 a32 a33

now

multiply

EA =

1 0 0 a11 a12 a13
c 1 0 a21 a22 a23
0 0 1 a31 a32 a33

=
a11 a12 a13
ca11+ a21 ca12 +a22 ca13 + a23
a31 a32 a33


which is same as above matrix

now

E^-1 =

1 0 0
-c 1 0
0 0 1

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