Matrices and Matrix Algebra

1. Examples of Matrix Multiplication

Recall from the preceding lecture our definition of matrix multiplication.

Definition 4.1. Let A be an m by n matrix and let B be an s by t matrix. If n ≠ s the matrix product
AB is not defined (i.e. if the number of columns of A does not equal the numbe of rows of B, the matrix
product is not defined ). If n = s, then the matrix product AB is defined and is the m by t matrix whose
entries (AB)_ij are prescribed by

In other words, the entry in j^th column of the i^th row of the product matrix AB is the dot product the vector
correspondind to the i^th row of A and the vector corresponding to the j^th column of B.

Let’s now compute some illustrative examples

Example 4.2.

does not exist

Because we need the same number of columns in the first factor as there are rows in the second factor.
Example 4.3.

So even though the 2 by 1 matrix and the 1 by 2 matrixcorrespond to the same
2-dimensional vector (1, −1), their products with the 2 by 2 matrix are not the same.
Example 4.4.

So the product AB of two matrices A and B is not necessarily the same as the product BA. In other
words, matrix multiplication is not commutative in general. Indeed, it can happen that AB exists but BA
is not even defined.

Note that this circumstances partially explains the paradox of the first example. Let A denote the 2 by 2
matrix If we interprete the vector (1,−1) as a 2 by 1 matrix v , then only the product Av is
defined; and if we interprete the vector (1,−1) as a 1 by 2 matrix then only the product vA is defined
Example 4.5.

Recall that for real numbers x^2 = 0 implies x = 0. This is evidently not the case for matrices: it can happen
that A^2 = 0 but A is not equal to the zero matrix 0.

Example 4.6.

Recall that for real numbers xy = 0 implies either x = 0 or y = 0. This is evidently not the case for
matrices: it can happen that AB = 0 but neither A or B is equal to the zero matrix 0.

Example 4.7.

And so muliplying any 3 by 3 matrix A by the matrix

just replicates the matrix A:

AI = IA = A

The example above generalizes to arbitrary n by n matrices (i.e. “ square matrices ”). This motivates the
fol lowing definition .

Definition 4.8. Let I be the n by n matrix whose entries are given by

In other words, I is an n by n matrix with 1’s along the diagonal (running from the upper left to the lower
right) and 0’s everywhere else. We call such a matrix the n by n identity matrix. It has the property that
IA = AI = A for all n by n matrices A except the 0 matrix.

2. Other Matrix Ope rations

Definition 4.9. Let A be an m by n matrix, and let r be any real number. Then the scalar product rA is
defined as the m by n matrix whose ij^th entry is r times the ij^th entry of A:

Example 4.10. If

then

Definition 4.11. Let A and B be m by n matrices. Then the matrix sum A +B is defined as the m by n
matrix whose ij^th entry is the sum of the ij^th entries of A and B:

Example 4.12. If

then

Combining these two operations of scalar multiplication and addition we can now from linear combinations
of matrices; e.g. 2A−3B.

Definition 4.13. Let A be an m by n matrix, then the transpose A^T of A is the n by m such that

In other words, the entries in the i^th row of A^T are identical to the entries in the i^th column of A.
Example 4.14. If

then

Example 4.15. Recall that we can interprete an n-dimenional either as a n by 1 matrix
(which we called a column vector)

or as a 1 by n matrix (which we called a row vector)

Note that

and

Definition 4.16. An n by n matrix with the property that A = A^T is called a symmetric matrix.

Example 4.17.

is a symmetric matrix, but

is not symmetric because, for example

With a little experience it is easy to glance a matrix and de termine whether or not it’s symmetric.
Theorem 4.18. Suppose the matrix product AB is defined, then