Monday, March 17 − Lecture 28 : Determinants
1. Define the (i, j)-minor of a square matrix.
2. Define the (i, j)-cofactor of a square matrix.
3. Calculate the determinant of a square matrix by row or column expansion.
4. Apply the fact that the determinant of a triangular matrix is the product of the
entries on the diagonal.
5. Recognize that a triangular matrix A is invertible iff its determinant is not 0.
28.1 Determinants − Let M denote the set of all square matrices. A “determinant” can be
viewed as a function D : M → ℝ which maps any n × n matrix A to a real number
according to well defined formula applied to the entries of the matrix. Its general formula
will be stated explicitly below. We denote the determinant function by det(A) or | A |.
In the case of a 2 × 2 matrix A = [a ] ij 2 × 2the function “det” is defined as:
det(A) = a a 11 22 . 12 21
28.2 Definition − Let A = [a ] be an ij× n matrix. For each entry a in A, we associateij
matrix M of ijmension n − 1 by n − 1 whose entries are the ones that remain when we
remove from A the row and the column containing the entry a . We will refer to ijas a ij
minor matrix of A associated to a ij We will call the number
m ijdet(M ) = |ij | ij
the minor of the elementa , or the (i, j) minor.
Remark − Note that M is a matijx while m is a real numbeij Also note that we have not
yet defined what det(M ). ij
Second Remark − For an n × n matrix A there are n minors m , one for each entryijf A.
28.3 Definition − For each minor m of an n × nijatrix A, we define the (i, j)-cofactor c as ij
i + j
c ij(−1) m .ij
28.3.1 Example. Let Abe the matrix Then (2, 3)-minor matrix, M , is
2 + 3
Then m =23et(M ) = 23– 0 = 3 and c = (–1)23 m 23–3.
28.4 Definition − For an n × n matrix A = [a ] ij n × nSuppose the det(B m × m) is defined for
any positive integer m < n. Then all (i, j)-cofactors of A are all well-defined numbers.
We define the determinant of A, det(A) = | A | as :
Remark – Let M denote the set of all square matrices and M denote the set of all
n × n
n × n matrices for some n. According to this definition we see that the determinant
function D : M → ℝ is a “recursively defined function” on the set of positive integers.
This means that we cannot determine the values of the determinants D: M n × n → ℝ
unless we know the values of the determinants in D : M m × m→ ℝ for all m < n. It can be
proven that such recursively defined functions are well-defined functions on the set of all
square matrices M. Having acknowledged this, we will henceforth accept this as a
Remark – We will soon see that there are many(often more convenient) ways of
computing the value of a determinant det(A).
28.4.1 Example – Let A be the matrix
We begin the “cofactor expansion along row 1” allowing us to find det A:
We compute det M : 14 So
We allow the reader to verify in this way the values of det M 11, det M 12d det M . 13
By proceeding in this ways for the other minors we get det(A) = −256.
28.4.2 Remark − This method of evaluating a determinant of A is referred as “the
cofactor expansion of det(A) along the first row”.
28.5 Theorem − For a square n × n matrix A = [a ] ij n × ne determinant det(A) can also be