7. Let G be the set of all 2 × 2 matrices of the...

7. Let G be the set of all 2 × 2 matrices of the form egin{pmatrix} a & b \ -b & a end{pmatrix} where a, b ? ? and a² + b² ? 0. Show that G forms a group under matrix multiplication.

Transcript

00:01 Hello everyone, in this problem we are given with j be a set of all 2x2 matrices of the form as a, b, –b and a where ab belongs to r and a2 plus b2 does not equals to 0.

00:26 Now we need to show that j forms a group under matrix multiplication.

00:32 So, in order to prove that first let us begin with closure property.

00:37 So, for closure property let us take a be a matrix of ab, –b and a and b to be a ' b ' –b ' and a'.

00:53 So, for these 2 matrices are belongs to the group.

01:00 Now we need to prove that ab belongs to g.

01:06 So, for that let us multiply the matrices a and b.

01:11 So, we have that value to be ab – ba multiplied by a ' b ' – b ' and a'.

01:22 By performing matrix multiplication we have the values to be by multiplying first row with first column.

01:33 So, we have aa ' – b b ' and first row with the second column ab ' – b ' – b ' and second row with the first column –ba ' – ab ' and second row with the second column it would be –bb ' plus aa'.

01:57 So, now we can rewrite this values of this matrix to be aa ' –bb ' ab ' plus ba ' minus of ab ' plus ba ' and aa ' minus b b'.

02:16 So, now we can take this value to be c this value to be d.

02:21 So, it would become here it is of minus d and it is of c.

02:25 So, which is in the math which is the same as in the form of g.

02:29 So, this belongs to g.

02:31 So, we have proved that the product of a and b belongs to g.

02:36 So, now the closure property is verified.

02:41 Now, let us move on to associative property.

02:44 Associative.

02:49 So, now let us take abc belongs to g then a of bc to be equal to ab of c...

Question

7. Let G be the set of all 2 × 2 matrices of the form egin{pmatrix} a & b \ -b & a end{pmatrix} where a, b ? ? and a² + b² ? 0. Show that G forms a group under matrix multiplication.

Please give Ace some feedback

7. Let G be the set of all 2 × 2 matrices of the form egin{pmatrix} a & b \ -b & a end{pmatrix} where a, b ? ? and a² + b² ? 0. Show that G forms a group under matrix multiplication.