(b) Consider a linear system Ax = b whose augmented matrix (A b) is a 6 xn matrix of rank k. where the natural numbers n and k satisfy n> 2 and k > 2, Answer the following questions, carefully explaining each answer. ) If k 5 and the system Ax b is inconsistent, what are the possible values that n can take? (i) If A take? 4 and the system Ax- b is consistent, what are the possible values that n can (iii) If A= 3 and the system Ax bhas infinitely many solutions, what are the possible values that n can take?

Solve for x,y,z and w in the matrix equation wxyx=-432-1+2ywzx.

What operations can be performed on a linear system to arrive at an equivalent system?

What is the reduced row echelon form of a matrix?

What is the row-echelon form of a matrix? What is a leading entry?

A factory manufactures three products (doohickies, gizmos, and widgets) and ships them to two warehouses for storage. The number of units of each product shipped to each warehouse is given by the matrix A=[20015010075100125] (where aij is the number of units of product i sent to warehouse j and the products are taken in alphabetical order). The cost of shipping one unit of each product by truck is $1.50 per doohickey, $1.00 per gizmo, and $2.00 per widget. The corresponding unit costs to ship by train are $1.75, $1.50, and $1.00. Organize these costs into a matrix B and then use matrix multiplication to show how the factory can compare the cost of shipping its products to each of the two warehouses by truck and by train.

Show that no 22 matrices A and B exist that satisfy the matrix equation. AB-BA=1001.

A manufacture produce three models of a product and ships them to two warehouse. In the matrix A=[200300600350250400] aij represent the number of units model i that the manufacturer ships to warehouse j. The matrix B=[12.509.0021.50] Represent the price of the three models in dollars per unit. Find the product BA and state what each entry of the matrix represents.

If A and B are matrices with the same dimension and k is a real number, how do you find A+B and kA ?