Is T a group with respect to matrix multiplication? explain

Use the standard matrix for counterclockwise rotation in R2 to rotate the triangle with vertices (3,5), (5,3) and (3,0) counterclockwise 90 about the origin. Graph the triangles.

Define Matrix Notation.

Let SL₂(R)= {A|A is a 2x 2 matrix with det(A)=1}. We will show that SL₂(R) is a group with matrix multiplication: (a) Let A,BE SL₂(R). Find det(AB). (b) IS AB E SL₂(R) ? O No, SL₂(R) is not closed under multiplication. Yes, SL₂(R) is closed under multiplication. (c) I= We already know that matrix multiplication is associative and that 0 is a multiplicative identity for matrix multiplication. Find det(I). 01 (d) Is IE SL₂(R) ? O No, SL₂(R) does not contain an identity element. O Yes, SL₂(R) contains an identity element. (e) Let AE SL₂(R). Find det(A-¹). (f) Is A-¹ ESL₂(R) ? O Yes, every element of SL2(R) has an inverse in SL₂(R). O No, not every element of SL₂(R) has an inverse in SL₂(R).

It can be shown that A and B are elements of the group of 2 x 2 invertible matrices with real entries under the binary operation of matrix multiplication. Show 3. Let A = and B = that A and B have finite orders but AB does not. Recall that the order of an element in a group is the number of times that it must be combined with itself in order to produce the identity element.

Prove that the set of all 2 x 2 matrices with entries from R and determinant+1 is a group under matrix multiplication.

both b anc c if possible

If f and g are linear transformations are represented by the matrices (0 -4/2 1) and (3 3/-5 -4), what is the image of the point (-1,1) under the composite transformation g o f?

Determine whether or not ( ABC )" = C"B"A" for matrices: A, B, C

2) Consider the set SL(n, R) consisting of n x n matrices with real entries having de- terminant equal to 1. Prove that SL(n, R) is a group under the operation of matrix multiplication (it is referred to as the Special Linear Group).