8.2 Elementary Matrices and Determinants
8.2.4 Determinant of Products
In summary, the elementary matrices for each of the row operations obey
Ei
j = I with rowsi, j swapped; detEji =−1
Ri(λ) = I with λ in positioni, i; detRi(λ) = λ
Sji(µ) = I with µin positioni, j; detSji(µ) = 1
Elementary Determinants
Moreover we found a useful formula for determinants of products: Theorem 8.2.1. If E is any of the elementary matrices Eji, Ri(λ), Sji(µ), then det(EM) = detEdetM.
We have seen that any matrixM can be put into reduced row echelon form via a sequence of row operations, and we have seen that any row operation can be achieved via left matrix multiplication by an elementary matrix. Suppose that RREF(M) is the reduced row echelon form of M. Then
RREF(M) = E1E2· · ·EkM ,
where eachEi is an elementary matrix. We know how to compute determi- nants of elementary matrices and products thereof, so we ask:
What is the determinant of a square matrix in reduced row echelon form? The answer has two cases:
1. IfM is not invertible, then some row of RREF(M) contains only zeros. Then we can multiply the zero row by any constant λ without chang- ing M; by our previous observation, this scales the determinant of M
by λ. Thus, if M is not invertible, det RREF(M) = λdet RREF(M), and so det RREF(M) = 0.
2. Otherwise, every row of RREF(M) has a pivot on the diagonal; since
M is square, this means that RREF(M) is the identity matrix. So if
M is invertible, det RREF(M) = 1.
Notice that because det RREF(M) = det(E1E2· · ·EkM), by the theorem above,
det RREF(M) = det(E1)· · ·det(Ek) detM .
Since eachEi has non-zero determinant, then det RREF(M) = 0 if and only if detM = 0. This establishes an important theorem:
Theorem 8.2.2. For any square matrix M, detM 6= 0 if and only if M is invertible.
Since we know the determinants of the elementary matrices, we can im- mediately obtain the following:
8.2 Elementary Matrices and Determinants 181
Figure 8.5: Determinants measure if a matrix is invertible. Corollary 8.2.3. Any elementary matrix Ei
j, Ri(λ), Sji(µ) is invertible, ex-
cept for Ri(0). In fact, the inverse of an elementary matrix is another ele- mentary matrix.
To obtain one last important result, suppose that M and N are square
n ×n matrices, with reduced row echelon forms such that, for elementary matrices Ei and Fi,
M =E1E2· · ·Ek RREF(M), and
N =F1F2· · ·Fl RREF(N).
If RREF(M) is the identity matrix (i.e., M is invertible), then:
det(M N) = det(E1E2· · ·Ek RREF(M)F1F2· · ·Fl RREF(N)) = det(E1E2· · ·EkIF1F2· · ·Fl RREF(N))
= det(E1)· · ·det(Ek) det(I) det(F1)· · ·det(Fl) det RREF(N) = det(M) det(N)
Otherwise, M is not invertible, and detM = 0 = det RREF(M). Then there exists a row of zeros in RREF(M), so Rn(λ) RREF(M) = RREF(M) for
any λ. Then:
det(M N) = det(E1E2· · ·Ek RREF(M)N)
= det(E1)· · ·det(Ek) det(RREF(M)N)
= det(E1)· · ·det(Ek) det(Rn(λ) RREF(M)N) = det(E1)· · ·det(Ek)λdet(RREF(M)N) = λdet(M N)
Figure 8.6: “The determinant of a product is the product of determinants.” Which implies that det(M N) = 0 = detMdetN.
Thus we have shown that for any matrices M and N, det(M N) = detMdetN
This result isextremely important; do not forget it!
Alternative proof
Reading homework: problem 4
8.3
Review Problems
Webwork:
Reading Problems 1 , 2 , 3 , 4
2×2 Determinant 7
Determinants and invertibility 8, 9,10, 11
1. Let M = m1 1 m12 m13 m2 1 m22 m23 m3 1 m32 m33 .
8.3 Review Problems 183
Use row operations to put M into row echelon form. For simplicity, assume that m1
1 6= 0 6=m11m22−m21m12. Prove that M is non-singular if and only if:
m11m22m33−m11m32m32+m12m23m31−m12m21m33+m13m12m32−m13m22m31 6= 0 2. (a) What does the matrix E21 =
0 1 1 0 do to M = a b d c under left multiplication? What about right multiplication?
(b) Find elementary matricesR1(λ) andR2(λ) that respectively mul- tiply rows 1 and 2 of M by λ but otherwise leave M the same under left multiplication.
(c) Find a matrix S21(λ) that adds a multiple λ of row 2 to row 1 under left multiplication.
3. Let ˆσ denote the permutation obtained from σby transposing the first two outputs, i.e. σˆ(1) = σ(2) and ˆσ(2) = σ(1). Suppose the function
f :{1,2,3,4} →R. Write out explicitly the following two sums: X
σ
f σ(s) and X σ
f σˆ(s).
What do you observe? Now write a brief explanation why the following equality holds X σ F(σ) = X σ F(ˆσ),
where the domain of the function F is the set of all permutations of n
objects and ˆσ is related to σ by swapping a given pair of objects. 4. Let M be a matrix and Si
jM the same matrix with rows i and j switched. Explain every line of the series of equations proving that detM =−det(Si
jM).
5. Let M0 be the matrix obtained from M by swapping two columns i
and j. Show that detM0 =−detM.
6. The scalar triple product of three vectorsu, v, w fromR3 isu·(v×w). Show that this product is the same as the determinant of the matrix whose columns are u, v, w (in that order). What happens to the scalar triple product when the factors are permuted?
7. Show that if M is a 3×3 matrix whose third row is a sum of multiples of the other rows (R3 = aR2 +bR1) then detM = 0. Show that the same is true if one of the columns is a sum of multiples of the others. 8. Calculate the determinant below by factoring the matrix into elemen-
tary matrices times simpler matrices and using the trick det(M) = det(E−1EM) = det(E−1) det(EM).
Explicitly show each ERO matrix.
det 2 1 0 4 3 1 2 2 2 9. Let M = a b c d and N = x y z w
. Compute the following: (a) detM.
(b) detN. (c) det(M N). (d) detMdetN.
(e) det(M−1) assuming ad−bc6= 0. (f) det(MT)
(g) det(M+N)−(detM+ detN). Is the determinant a linear trans- formation from square matrices to real numbers? Explain.
10. Suppose M =
a b c d
is invertible. Write M as a product of elemen- tary row matrices times RREF(M).
11. Find the inverses of each of the elementary matrices, Ei
j, Ri(λ), Sji(λ). Make sure to show that the elementary matrix times its inverse is ac- tually the identity.
12. Let ei
j denote the matrix with a 1 in the i-th row and j-th column and 0’s everywhere else, and let Abe an arbitrary 2×2 matrix. Com- pute det(A+tI2). What is the first order term (thet1 term)? Can you
8.3 Review Problems 185
express your results in terms of tr(A)? What about the first order term in det(A+tIn) for any arbitrary n×n matrix A in terms of tr(A)? Note that the result of det(A+tI2) is a polynomial in the variable t known as the characteristic polynomial.
13. (Directional) Derivative of the determinant:
Notice that det : Mnn → R (where Mnn is the vector space of all n×n
matrices) det is a function of n2 variables so we can take directional derivatives of det.
Let A be an arbitrary n×n matrix, and for all i and j compute the following: (a) lim t→0 det(I2+teij)−det(I2) t (b) lim t→0 det(I3+teij)−det(I3) t (c) lim t→0 det(In+teij)−det(In) t (d) lim t→0 det(In+At)−det(In) t
Note, these are the directional derivative in the ei
j and A directions. 14. How many functions are in the set
{f :{1, . . . , n} → {1, . . . , n}|f−1 exists}? What about the set
{1, . . . , n}{1,...,n}
?
Which of these two sets correspond to the set of all permutations of n