1. Let 12 be the 2 X 2 identity matrix. (a) [6 points] Is it true that A12 = A for all 2 x 2 matrices A? If so, give the numbered result from the lecture notes that tells you why it is true (e.g., \"Theorem 4.2\"). If not, give an example of a 2 x 2 matrix A where A12 a A. (b) [6 points] Is it true that AB = BA for all 2 x 2 matrices A and B? If so, give the numbered result from the lecture notes that tells you why it is true (e.g., \"Theorem 4.2\"). If not, give an example of 2 X 2 matrices A and B where AB 74 BA. (0) [6 points] Is it true that (AB)T = BTAT for all 2 x 2 matrices A and B? If so, give the numbered result from the lecture notes that tells you why it is true (e.g., \"Theorem 4.2\"). If not, give an example of 2 x 2 matrices A and B where (AB)T 4 BTAT. 2. Consider the variables 3:1, :52, 3:3. For each of the following systems, say whether or not it is a system of linear equations. . $1+$2w33:53 =0 (a) [4 pomts] { 2:31 _ $3 = 3 31131 = .2173 (b) [4 pOthS] 2.061 = $3 2.172 = $2 1I1(1)$1 + COS(2).'L'2 3133 = 0 3 (C) [4 points] { 225:1 5L'3 = e 3. Photosynthesis is the process by which plants use radiant solar energy to convert carbon dioxide 002 and water H20 into glucose 061171206 and oxygen gas 02. The chemical equation of the reaction has the form 331002 + $2H20 > $302 + 1340585205. (a) [5 points] Set up (but do not solve) a system of linear equations to balance the chemical equation. (b) [4 points] If LE4 = 1, then What must 3:1, 3:2, and 333 be? (Show your work.) Bonus I Score 2 extra points if you can explain mathematically why it is not possible for carbon dioxide 002 and hydrogen gas H2 alone to generate only glucose 05H 1205