Question 1. Suppose A is a 3 x 3 matrix. 1. Find a 3 x 3 matrix 31;\

Question 1. Suppose A is a 3 x 3 matrix. 1. Find a 3 x 3 matrix Sij so that the product SijA is the same as the matrix A after applying the row operation Ri+R,. 2. Find a 3 x 3 matrix Tij(c) so that the product is the same as the matrix A after applying the row operations Ri 4- cRJ Y Ri. 3. Find a 3 x 3 matrix Ui(c) so that the product is the same as the matrix A after applying the row operation cRi i Ri. 4. Row reduce (by hand) the matrix Carefully keep track of the row operations. 1 2 3 o 1 4 5 6 5. Now replicate your row reduction from Part 4 by using the matrices that you found in Parts 1, 2, and 3. For example, if the first row operation you perform is to interchange rows 2 and 3, then you would mul- tiply A on the left by S23 to get S23A. Do not evaluate the product, simply write out the matrices like 6. Compute the product of the row-operation matrices that you found in Part (5). Do not include A in your product. So in the example above, you would compute the matrix product Call this matrix B. 7. What is the matrix B? Explain your answer. Numeracy 2. As the CEO of Gilmore's Glorious Goods you have been allocated an operating budget for the year. The company consists of two executive departments, each department is run by three directors, each director has four project managers, and each project manager is working on five projects. If each project has precisely one project manager, and the budget is to be evenly divided amongst every project, what percentage of the budget does each project receive? Numeracy 3. Suppose we define a function on the natural numbers, where f (O) = 1 and f (n) = nf(n 1) for 1. Compute f (1), f (2), f (3), ... , f (10) (using a calculator as necessary). 2. Find a closed form expression for f (n); that is, a way of computing f (n) without having to compute f (n 1) Use this to compute f (100). 3. If n and k are natural numbers with k < n, write (n) (n l)(n 2) using only f and divison.