Mathematical Structure -- Matrices

Prerequisites:

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

An (n by k)-matrix is an array or table of numbers with n rows and k columns -- for example, the matrices below each have two rows and three columns.

Missing equation

Notice that we usually use capital letters to denote entire matrices and the corresponding small letter with two subscripts to denote the entries in a matrix. The first subscript denotes the row and the second subscript denotes the column. For example, the entry in the second row and third column of the matrix M above is

m₂₃ = 6

The set of (n by k) - matrices is a vector space with vector addition and scalar multiplication defined by

Missing equation

The zero vector in this vector space is the (n by k)-matrix with all zero entries.

Do each of the following calculations "by hand" and then check your answer in your CAS window.