Rules for Matrix Algebra

MA 2071

1. Matrix multiplication is, in general, not commutative. That is, in general, AB ≠ BA. What this means is that one should not change the order of matrices in an expression, unlike ordinary algebra. (see examples from class 3/17)

2. Matrix multiplication is linear. This is a very important concept! It amounts to , if c and k are scalars,

A(cX + kY) = cAX + k AY

in other words you can distribute products and you can factor constants out.

You have seen this concept in other places:

in Calculus I, derivatives are linear d(c f(x) + k g(x))= c df + kdg

dx dx dx

in Calculus II, integrals are linear

Also Fourier Transforms, important to Signal Processing, are linear

F( c(f(t) + kg(t) ) = c F (f(t)) + k F (g(t))

and Laplace Transforms, used in systems analysis, are linear

L( c(f(x) + kg(x) ) = c L(f(x)) + k L(g(x))

3. Matrix algebra is a binary operation. This means that at any time, no matter how many matrices are involved

in an expression, you are only multiplying or adding two at a time. You have your choice of which two, as long as

you do not reverse the order of multiplication. This is called the associative property:

(AB)C = A(BC) = ABC

4. The identity matrix, I_n, (1s on diagonal, 0s elsewhere) serves as the multiplicative identity:

AI_n = A and I_nA = A

(so this is an example of a rare time when order does not matter).

5. Some , but not all, square matrices have multiplicative inverses. This means another matrix which satisfies

AB = BA = I_n

In such a case, B is denoted by A^-1. If a matrix has an inverse, it is called nonsingular in the text book.

Lots of matrices do not have inverses, unlike ordinary numbers where only 0 has no multiplicative inverse.

Those that do not are called singular matrices.