Linear Algebra for Beginners
A matrix is a rectangular array of numbers and the numbers in the array are called the entries in the matrix. A two-dimensional matrix with one dimension equal to 1 is sometimes called a row matrix (a matrix with only one row) or a column matrix (a matrix with only one column). A matrix with both dimensions equal to 1 is simply a single number which we can also call a scalar. It is conventional to denote matrices by boldface upper-case symbols and row or column vectors by lower-case symbols. So, for example, the matrix A, where:
is a 2 × 2 matrix with four entries. Since only the diagonal entries are non-zero we can state that A is a diagonal matrix (furthermore, a diagonal matrix whose diagonal entries are all 1 is also called an identity matrix).
Two matrices are defined to be equal if they have the same size and their corresponding entries are equal. If A and B are matrices of the same size, then the sum A + B is the matrix obtained by adding the entries of B to the corresponding entries of A, and the difference A − B is the matrix obtained by subtracting the entries of B from the corresponding entries of A. Only matrices of the same size can be added or subtracted. As an example, if we defined matrix B by:
then we can write that:
If A is an m × r matrix and B is an r × n matrix, then ...