Math1300:MainPage/CofactorExpansion
Contents
• 1 The Determinant of a Matrix: Introduction
• 2 The Definition of Minors and Cofactors
• 3 Cofactor Expansions on Rows and Columns of a Matrix
• 4 The Definition of the Determinant Using the Cofactor Expansion Theorem
♦ 4.1 Theorem (Expansion by Cofactors)
♦ 4.2 Application of Cofactor Expansion to Small Matrices
♦ 4.3 Application of Cofactor Expansion to Triangular Matrices
• 5 Some Properties of Determinants Derived from Cofactor Expansion
♦ 5.1 Proposition (all zero row or column implies zero determinant)
♦ 5.2 Lemma (Interchange Rows One and Two)
♦ 5.3 Lemma (Interchange Two Consecutive Rows)
♦ 5.4 Proposition (Interchange Two Rows Changes the Sign of the Determinant)
♦ 5.5 Theorem (Interchange Two Rows or Columns Changes the Sign of the Determinant)
♦ 5.6 Proposition (Two Equal Rows or Columns Implies Zero Determinant)
♦ 5.7 Proposition (Multiply a Row or Column by a Constant Multiplies the Determinant by
the Constant)
♦ 5.8 Proposition (Multiply a Matrix by a Constant)
♦ 5.9 Proposition (Two Proportional Rows or Columns Implies Zero Determinant)
♦ 5.10 Proposition (Row Additivity)
The Determinant of a Matrix: Introduction
Each square matrix of order n has a number associated with it called its determinant. If the matrix is A we denote
its determinant by det(A).
• If n = 1, and A = [a1,1] then det(A) = a1,1.
•
If n = 2 and
, then det(A) = a a
−a a .
•
1,1 2,2
If n = 3 and
a1,1a2,3a3,2 − a1,2a2,1a3,3 − a1,3a2,2a3,1.
1,2 2,1
, then det(A) = a a a
+a a a
+a a a
−
Examples:
1,1 2,2 3,3
1,2 2,3 3,1
1,3 2,1 3,2
•
•
There is a theoretical formula for the evaluation of the determinant for larger n, but the number of summands
grows quickly (for n = 4 there are 24 terms to add, and for n = 5 there are 120). The present goal is to find better
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methods to evaluate the determinant. The first technique involves cofactors.
The Definition of Minors and Cofactors
The evaluation of the determinant by cofactors changes the problem of finding the determinant of a matrix of
order n to that of finding the determinant several matrices of order n − 1. So far, we know how to evaluate the
determinant of a matrix of order n where
This method will, then, allow us to evaluate the determinant of
a matrix of order n = 4 by reducing the problem to solving the determinant of several matrices of order n = 3.
Suppose we are given a matrix A of order n, and ai,j is one of its entries. The i-j minor Mi,j is defined to be the
determinant of the matrix obtained by deleting the row Ri and the column Cj from the matrix. The i-j cofactor Ci,j
= ( − 1)i + jMi,j.
Observe that
This means that
$-$.
Example: Find M
where the evenness or oddness of $i+j$ determines whether $\pm$ is $+$ or
and C
for the matrix
C from A we get the matrix
2,35
so M2,3 = 4 and C2,3
2,3 = ( − 1) 4 = − 4.
When we delete row R and column
We have already calculated the determinant of this matrix to be 4,
2
There is a nice way of visualizing the pattern of the sign of ( − 1)i + j. There is a checkerboard pattern of those
3
entries
with a + sign and those with a − sign.
For any matrix A, typical summand of the row expansion has three factors: ( − 1)i + jai,jMi,j where Mi,j is the
determinant of the * part of the matrix:
The Determinant of a Matrix: Introduction
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Cofactor Expansions on Rows and Columns of a Matrix
The evaluation of the determinant uses the concept of cofactor expansion along a row or column. Consider the
row Ri of a matrix A. The entries in Ri are
and these entries are used in the cofactor
expansion on that row:
Similarly, the cofactor expansion on column Cj is
Example:
Here is the cofactor expansion on the first row:
a1,1 = 1
C1,1 = ( − 1)2M1,1 = − 14
a1,2 = 2
C1,2 = ( − 1)3M1,2 = 36
a1,3 = 2
C1,3 = ( − 1)4M1,3 = − 16
The Definition of Minors and Cofactors
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a1,4 = 3
C1,4 = ( − 1)5M1,4 = − 26
Now we can compute
The cofactor expansion on column C3 is
a1,3C1,3 + a2,3C2,3 + a3,3C3,3 + a4,3C4,3 = a1,3M1,3 − a2,3M2,3 + a3,3M3,3 − a4,3M4,3
and so a1,3M1,3 − a2,3M2,3 + a3,3M3,3 − a4,3M4,3 = 2( − 16) − 5(4) + 8( − 12) − 2( − 48) = − 52
The cofactor expansion on column C4 evaluates to − 3(26) + 3(0) + 1(0) + 1(26) = − 52.
The three cofactor expansions of A give the identical result. This is an indicator of the next theorem.
The Definition of the Determinant Using the Cofactor
Expansion Theorem
Theorem (Expansion by Cofactors)
Let A by a square matrix of order n. Then the 2n cofactor expansions on the n rows
columns
are all identical.
and n
Proof: The proof is rather complicated and is postponed until this section.
For a square matrix A, the common value of the row and column expansions is called the determinant of A and is
denoted det(A).
An implication of this result is that any row or column expansion may be used to evaluate the determinant of a
matrix. Which is best? Here is a hint.
Cofactor Expansions on Rows and Columns of a Matrix
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Suppose we want to evaluate the determinant of the matrix
Notice that the expansion on the first row is 2M1,1 − 0M1,2 + 0M1,3 − 0M1,4 + 0M1,5 = 2M1,1.
Now
If we expand M
on its second column, we get
The determinant of this new matrix may be evaluated directly, or it can be
1,1
evaluated by a third row expansion. In either case we get
know det(A) = 8.
With this information, we
The moral of this evaluation: When choosing a row or column for cofactor expansion, zeros are your friends.
Application of Cofactor Expansion to Small Matrices
We wish to evaluate the determinant of square matrices of order n for
• n = 1: by definition det([a1,1]) = a1,1
•
n = 2: Take
a1,2det([a2,1]) = a1,1a2,2 − a1,2a2,1.
and expand on the first row. Then det(A) = a det([a ]) −
•
1,1
n = 3: Take
Theorem (Expansion by Cofactors)
2,2
and expand on the first row. Then
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We can visualize these determinants by looking at the diagonals:
For n = 2,
and the determinant is the product of the two diagonal elements minus the
product of the other two elements.
For n = 3, we may take A and append a copy of the first two columns to the right side. Then the determinant is the
sum of six terms: the three positive ones are the product of terms on the upper-left to lower-right diagonals while
the three negative ones re the product of terms on the lower-left to upper-right diagonals.
(positive terms are in same colour)
(negative terms are in same colour)
So it's easy to compute the determinant of a matrix of order n when
reasonable generalization for larger n.
Unfortunately, there is no
Application of Cofactor Expansion to Triangular Matrices
The determinant of a triangular matrix is the product of its diagonal elements.
Proof: Suppose that A is upper triangular.
The cofactor expansion on the last row gives det(A) = an,nMn,n where
Application of Cofactor Expansion to Small Matrices
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This new matrix is also upper triangular, so we may carry out the identical process on this matrix. After repeated
application,
The lower triangular proof uses the same argument
applied to the last column.
Some Properties of Determinants Derived from Cofactor
Expansion
The cofactor expansion theorem turns out to be a powerful tool, both for computation and for the derivation of
theoretical results. In this section we derive several of these results.
Proposition (all zero row or column implies zero determinant)
If A is a square matrix of order n with at all-zero row or column, then det(A) = 0.
Proof: Suppose
Then the cofactor expansion on that row is
and so det(A) = 0.
Lemma (Interchange Rows One and Two)
Let A be a square matrix of order n and B be the matrix obtained by interchanging the first and second rows of A.
Then det(B) = − det(A).
Proof For the matrix A let Ci,j be the cofactors and Mi,j be the minors, and for B let Di,j be the cofactors and Ni,j be
the minors. The definition of B implies that b1,j = a2,j and b2,j = a1,j.
Consider the first row expansion of A and the second row expansion of B. Note that deleting the first row of A is
exactly the same as deleting the second row of B. As such, M = N for
1,j
Application of Cofactor Expansion to Triangular Matrices
Then
2,j
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Lemma (Interchange Two Consecutive Rows)
Let A be a square matrix of order n and B be the matrix obtained by interchanging Ri and Ri + 1 of A. Then det(B) =
− det(A).
Proof The proof here is just that of the previous Lemma replacing of 1 by i and 2 by i + 1. As in the previous
Lemma, let Ci,j be the cofactors and Mi,j be the minors of A, and let Di,j be the cofactors and Ni,j be the minors for
B . Expand by cofactors row i of A and row i + 1 of B. Then ai,j = bi + 1,j and Mi,j = Ni + 1,j.
and
Proposition (Interchange Two Rows Changes the Sign of the Determinant)
Let A be a square matrix of order n and B be the matrix obtained by interchanging Ri and Rj Then det(B) = −
det(A).
Proof With no loss of generality we may assume that i < j. First, take row Ri and repeatedly interchange it with
the one below it until it is just below Rj. This takes j − i interchanges. Now take row Rj and interchange it with the
one above it until it is in the original position of Ri. This takes j − i − 1 interchanges. The effect is to interchange
row Ri and row Rj, that is, the result is B. Since the total number of interchanges is 2(j − i) − 1 and each
interchange causes the determinant to change sign, det(B) = ( − 1)2(j − i) − 1det(A) = − det(A).
Theorem (Interchange Two Rows or Columns Changes the Sign of the Determinant)
Suppose B is obtained from A by interchanging two rows or two columns. Then det(A) = − det(B).
Proof Apply the previous row expansion arguments to column expansion.
Proposition (Two Equal Rows or Columns Implies Zero Determinant)
If a square matrix A has two equal rows or columns, then det(A) = 0.
Proof Interchange the two equal rows or columns. This leaves the matrix unchanged but changes the sign of the
determinant, so det(A) = − det(A) and hence det(A) = 0.
Lemma (Interchange Rows One and Two)
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Proposition (Multiply a Row or Column by a Constant Multiplies the Determinant by the
Constant)
Let B be obtained from A by replacing Ri with λRi. Then det(B) = λdet(A).
Proof: Note that the i-j cofactors of A and B are identical, and b = λa for
i,j
i,j
row Ri:
. Expand B along
Proposition (Multiply a Matrix by a Constant)
Let A be a square matrix of order n. Then det(λA) = λndet(A).
Proof Take A and replace R by λR for
by λ. The resulting matrix isi λA. i
Each of the n replacements multiplies the determinant
Two rows R and R are proportional if R = λR for some
i
j
i
j
Proposition (Two Proportional Rows or Columns Implies Zero Determinant)
If a square matrix A has two proportional rows or columns, then det(A) = 0.
Proof: Apply
to A to get a new matrix B. We then know that det(B) = λdet(A) and, since B has
two identical rows, that det(B) = 0. This implies that det(A) = 0.
For the next result, we assume that A = [ai,j] and B = [bi,j] are given and are identical except for entries in row Ri.
In other words a = b if
Then C is the matrix where
A way to visualize these matrices:
r,s
r,s
Proposition (Multiply a Row or Column by a Constant Multiplies the Determinant by the Constant)
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Proposition (Row Additivity)
If A, B and C are defined as above, then det(C) = det(A) + det(B).
Proof: First note that a + b = c and the cofactors C are identical for all three matrices for
i,j
i,j
i,j
Expanding on row Ri: i,j
Proposition (Two Proportional Rows or Columns Implies Zero Determinant)
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