Exercise 1 Consider the matrices, 2 1 3 0 A=(5 o) B=(0 2) c=() d=(3 5 1) 0 3 1 1 calculate the following matrix quantities. You must show your work. Report your answer in simplest form. Highlight or box your answer. (a) The linear transformation, Ac = (b) The matrix product, AB = (c) The outer product, cd = (d) The inner product, dc = (e) The matrix transpose, AT = (f) The determinant, det (A) = (g) The matrix trace, T1- (A) = Note. The outer product of a column vector, u, and a row vector, v is just the matrix multiplication of a (m x 1) matrix, 11 with a (1 x 11) matrix, v. For example if u = (:1) v: (01 02) 2 u uv = 101 \"102 \"201 \"202 whereas the inner product vu, would be equivalent to the scalar dot product, then, vu = ulvl + H2212 Exercise 2 Consider the unbalanced combustion reaction, C2H6 + 02 - CO2 + H2O use Gaussian Elimination to balance the reaction. (a) Write a stoichiometric system of equations for the reaction. You will have 3 linear equations (one for each atom) and four unknowns (the coefficients on each of the four molecules in the reaction). (b) (b) From (a) you will have a homogeneous linear system. Write the augmented matrix for this system. It will have four rows and five columns. (c) Use Gaussian Elimination to solve the system of equations. You may either do this manually or use RowReduce in Mathematica to obtain you solution. You do not need to upload your Mathematica code if you use RowReduce. Report your answer as a matrix in reduced row echelon form. (d) From the matrix you obtained in (c), write the balanced reaction. You may multiply each coefficient by a scalar to make them whole numbers or leave the coefficients fractional