66 CHAPTER 2. LINEAR TRANSFORMATIONS

This theorem shows that if an n × n matrix B acts like an inverse when multiplied onone side of A, it follows that B = A−1and it will act like an inverse on both sides of A.

The conclusion of this theorem pertains to square matrices only. For example, let

A =

 1 0

0 1

1 0

 , B =

(1 0 0

1 1 −1

)(2.27)

Then

BA =

(1 0

0 1

)but

AB =

 1 0 0

1 1 −1

1 0 0

 .

2.8 Matrices and Calculus

The study of moving coordinate systems gives a non trivial example of the usefulness of theideas involving linear transformations and matrices. To begin with, here is the concept ofthe product rule extended to matrix multiplication.

Definition 2.8.1 Let A (t) be an m × n matrix. Say A (t) = (Aij (t)) . Suppose also thatAij (t) is a differentiable function for all i, j. Then define A′ (t) ≡

(A′

ij (t)). That is, A′ (t)

is the matrix which consists of replacing each entry by its derivative. Such an m×n matrixin which the entries are differentiable functions is called a differentiable matrix.

The next lemma is just a version of the product rule.

Lemma 2.8.2 Let A (t) be an m × n matrix and let B (t) be an n × p matrix with theproperty that all the entries of these matrices are differentiable functions. Then

(A (t)B (t))′= A′ (t)B (t) +A (t)B′ (t) .

Proof: This is like the usual proof one sees in a calculus course.

1

h(A (t+ h)B (t+ h)−A (t)B (t)) =

1

h(A (t+ h)B (t+ h)−A (t+ h)B (t)) +

1

h(A (t+ h)B (t)−A (t)B (t))

= A (t+ h)B (t+ h)−B (t)

h+A (t+ h)−A (t)

hB (t)

and now, using the fact that the entries of the matrices are all differentiable, one can passto a limit in both sides as h→ 0 and conclude that

(A (t)B (t))′= A′ (t)B (t) +A (t)B′ (t)■