Page 240 - Linear Algebra Done Right

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Determinant of a Matrix
Our goal is to prove that det T = det M(T) for every T ∈L(V) and
every basis of V. To do this, we will need to develop some proper-
be devoted just to
deriving properties of
ties of determinants of matrices. The lemma below is the first of the An entire book could
properties we will need. determinants.
Fortunately we need
10.28 Lemma: Suppose A is a square matrix. If B is the matrix only a few of the basic
obtained from A by interchanging two columns, then properties.
det A =− det B.
Proof: Suppose A is given by 10.24 and B is obtained from A by
interchanging two columns. Think of the sum 10.25 defining det A and
the corresponding sum defining det B. The same products of a’s appear
in both sums, though they correspond to different permutations. The
permutation corresponding to a given product of a’s when computing
det B is obtained by interchanging two entries in the corresponding
permutation when computing det A, thus multiplying the sign of the
permutation by −1 (see 10.23). Hence det A =− det B.

If T ∈L(V) and the matrix of T (with respect to some basis) has two
equal columns, then T is not injective and hence det T = 0. Though
this comment makes the next lemma plausible, it cannot be used in the
proof because we do not yet know that det T = det M(T).

10.29 Lemma: If A is a square matrix that has two equal columns,
then det A = 0.

Proof: Suppose A is a square matrix that has two equal columns.
Interchanging the two equal columns of A gives the original matrix A.
Thus from 10.28 (with B = A), we have det A =− det A, which implies
that det A = 0.
This section is long, so let’s pause for a paragraph. The symbols✽

that appear on the first page of each chapter are decorations intended
to take up space so that the first section of the chapter can start on the
next page. Chapter 1 has one of these symbols, Chapter 2 has two of
them, and so on. The symbols get smaller with each chapter. What you
may not have noticed is that the sum of the areas of the symbols at the
beginning of each chapter is the same for all chapters. For example, the
√
diameter of each symbol at the beginning of Chapter 10 equals 1/ 10
times the diameter of the symbol in Chapter 1.