*Algebra*

## Miscellaneous Problems for Chapter 2

Problem 2.M.1 Suppose \(M\) is an invertible 2×2 integer matrix whose inverse is also an integer matrix. Then \(\det M\) and \(\det(M^{-1})\) are both integers. But they are also reciprocals of each other. This is only possible when \(\det M = \pm 1\). This condition is not only necessary but also sufficient—formula (1.1.17) for the inverse of a 2×2 matrix implies that whenever \(M\) is an integer matrix with determinant \(\pm 1\), \(M^{-1}\) will be an integer matrix too.

Now suppose \(a, c \in \mathbb{Z}\) with \(\gcd(a, c) = 1\). By
Corollary 2.3.6, there always exist \(b, d \in
\mathbb{Z}\) such that \(ad - bc = 1\), so \((a\ c)^t\) can occur as the
first column of some integer matrix with an integer inverse matrix. Conversely,
if \(a = c = 0\) then obviously a matrix with first column \((a\ c)^t\) is
singular, and if \(\gcd(a, c) > 1\) then the determinant of any integer
matrix with first column \((a\ c)^t\) will also be a multiple of that GCD. So
the answer to the problem is precisely those \(a, c\) that are relatively
prime

.

Problem 2.M.2

- A group of even order can be partitioned into equivalence classes by the relation defined so that \(a \sim b\) iff \(a = b\) or \(a = b^{-1}\). Then the number of such equivalence classes of odd size must be even. The element 1 is in its own equivalence class, so there must be some other equivalence class of odd size. Obviously every equivalence class has size 1 or 2. So there is some element \(a \neq 1\) that is in an equivalence class containing only itself, that is, \(a = a^{-1}\). Therefore \(a\) has order 2.
- The argument used in the solution to Exercise 2.8.4 proves the existence of an element of order 3 (however, it cannot be used to prove the existence of an element of order 7).

Problem 2.M.3 If \(G\) contains an element of order 6 then it is isomorphic to \(C_6\). If \(G\) contains an element of order 3 (call it \(x\)) but no element of order 6, then by Problem 2.M.2(a), it must contain an element of order 2 as well (call it \(y\)), and by an argument given in the solution to Exercise 2.11.6, the six elements \(1, x, x^2, y, xy, x^2y\) must be distinct. It is easy to see that \(yx \notin \{1, x, x^2, y\}\), so either \(yx = xy\) or \(yx = x^2 y\). In the former were true, then \((xy)^n = x^n y^n\) and \(xy\) would have order 6 (a contradiction). In the latter case, \(G\) would be isomorphic to \(S_3\). Finally, the last possibility, where all elements of \(G\) have order 1 or 2, can't occur. For in such a group we could choose two arbitrary elements \(x, y \in G \setminus \{1\}\) with \(x \neq y\), which are their own inverses, so \(xy = (xy)^{-1} = y^{-1}x^{-1} = yx\), implying that \(x\) and \(y\) generate a subgroup of order 4, which violates Lagrange's theorem. We conclude that all groups of order 6 are isomorphic to either \(C_6\) or \(S_3\).

Problem 2.M.5 All we need to prove is that each element of \(S\) has an inverse. Let \(a \in S\). Then consider the map \(L_a : S \to S\) given by \(L_a(b) = ab\). The Cancellation Law states that this map is injective. Since \(S\) is finite, \(L_a\) is also surjective, that is, 1 lies in its image. The \(b\) for which \(L_a(b) = 1\) is then a right inverse of \(a\). By considering the map \(R_a : S \to S\) with \(R_a(b) = ba\), we can similarly deduce the existence of a left inverse. We conclude that \(a\) is invertible for each \(a \in S\).

Problem 2.M.6

- Suppose \(a \in S\). Then the constant path \(X(t) = a\) lies within \(S\). So \(a \sim a\). Now suppose \(a \sim b\) using some path \(X\) lying in \(S\); then define \(X^r : [0, 1] \to \mathbb{R}^k\) by \(X^r(t) = X(1-t)\); then \(X^r(0) = b\), \(X^r(1) = a\), \(X\) is continuous, and \(X^r\) lies entirely in \(S\), so \(b \sim a\). Finally, suppose \(a \sim b\) and \(b \sim c\) using paths \(X\) and \(Y\) lying in \(S\). Then define \(Z : [0, 1] \to \mathbb{R}^k\) as follows: \[ Z(t) = \begin{cases} X(2t) & \text{if $t ≤ 1/2$} \\ Y(2t-1) & \text{if $t > 1/2$} \end{cases} \] Now the image of \(Z\) lies entirely in \(S\) since \(Z\) only takes on values that \(X\) and \(Y\) take on. The functions \(t \mapsto X(2t)\) and \(t \mapsto Y(2t-1)\) are continous since \(X\) and \(Y\) are; so \(Z\) is continuous provided that it is continous at \(t = 1/2\), which it is since \(X(1) = Y(0) = b\). Finally, \(Z(0) = X(0) = a\) and \(Z(1) = Y(1) = c\). So \(a \sim c\). The above results imply that \(\sim\) is an equivalence relation.
- This follows from Proposition 2.7.4.
- The first set is the unit circle, which is path-connected since there is an arc joining any pair of points on the circle. The second set is the union of the x and y axes, which is path-connected since every point \(P\) in the set is connected to the origin by the line segment with the origin and \(P\) as endpoints. The third set is not path-connected, since \((1, 1) \not\sim (-1, -1)\); to see this, observe that any continuous function taking on the values \((1, 1)\) and \((-1, -1)\) must pass through the y-axis (by applying the intermediate value theorem to its first component), but no point on the y-axis satisfies \(xy = 1\).

Problem 2.M.7

- Suppose \(X\) is a path from \(A\) to \(B\) that lies in \(G\), and \(Y\) is a path from \(C\) to \(D\) that lies in \(G\). Define \(Z : [0, 1] \to G\) by \(Z(t) = X(t)Y(t)\). Then \(Z\) is continuous since matrix multiplication is continuous, \(Z(0) = AC, Z(1) = BD\), and since each \(X(t)\) and \(Y(t)\) lies in \(G\), and \(G\) is a group, each \(X(t)Y(t)\) lies in \(G\) as well. Thus, \(Z\) is a path from \(AC\) to \(BD\) lying in \(G\).
Denote the set described in the problem by \(N\). We are given that \(I \in N\). Also, if \(A, B \in N\), then \(I \sim A\) and \(I \sim B\), and by part (a) we have \(I \sim AB\), so \(AB \in N\). Also, for every \(A \in N\), let \(X\) be a path from \(I\) to \(A\) that lies in \(G\); then define \(Y : [0, 1] \to G\) by \(Y(t) = X(t)^{-1}\). This only takes on values in \(G\) since \(X\) only takes on values in \(G\) and \(G\) is a group; it is continous since matrix inversion is continous; and also \(Y(0) = I\) and \(Y(1) = A^{-1}\). So \(I \sim A^{-1}\). These results taken together establish that \(N\) is a subgroup of \(G\).

Let \(A \in G\) and \(B \in N\). Using \(A \sim A\) and \(B \sim I\) and part (a), obtain that \(AB \sim A\). Using this together with \(A^{-1} \sim A^{-1}\) and part (a), obtain that \(ABA^{-1} \sim I\). Since this holds for all \(A, B\), conclude that \(N\) is normal in \(G\).

Problem 2.M.8

- We proved for Exercise 2.4.8(b) that the elementary matrices of the first type generate \(SL_n(\mathbb{R})\). Each elementary matrix of the first type can be joined to the identity matrix by a continous path that lies in \(SL_n(\mathbb{R})\) simply by contracting the entry \(a\) to zero. That is, \(I \sim E\) for every elementary matrix \(E\) of the first type. Suppose \(A \in SL_n(\mathbb{R})\). Write \(A = E_1 E_2 \ldots E_n\). Then since \(I \sim E_1, I \sim E_2, \ldots, I \sim E_n\), use 2.M.7(a) to conclude that \(I \sim A\). This holds for all \(A \in SL_n(\mathbb{R})\), so \(SL_n(\mathbb{R})\) is path-connected.
Suppose \(A, B \in GL_n(\mathbb{R})\) and \(\det A \det B < 0\). Then for any path \(X\) from \(A\) to \(B\), by the intermediate value theorem, the continous function \(\det \circ X\) must take on the value 0, therefore the image of \(X\) must contain a singular matrix. So \(A\) and \(B\) cannot be connected by a path in \(GL_n(\mathbb{R})\).

Suppose \(\det A > 0\) and \(\det B > 0\). Then consider the path \(X : [0, 1] \to GL_n(\mathbb{R})\) defined by \(X(t) = (\det A)^{-t/n} A\). This is continuous, and satisfies \(X(0) = A, X(1) = (\det A)^{-1/n}A\), so that \(\det X(1) = ((\det A)^{-1/n})^n \det A = 1\), that is, \(X(1) \in SL_n(\mathbb{R})\). Analogously \(B \sim (\det B)^{-1/n} B\) where the latter matrix is in \(SL_n(\mathbb{R})\). Since \(SL_n(\mathbb{R})\) is path-connected, the transitive property of path-connectivity implies that \(A \sim B\). If on the other hand \(\det A < 0\) and \(\det B < 0\), then let \(C = \diag(-1, 1, 1, \ldots, 1)\). We then have \(AC \sim BC\) since \(AC, BC\) both have positive determinant, and \(C^{-1} \sim C^{-1}\), so by 2.M.7(a), \(A \sim B\).

In conclusion, the two connected components of \(GL_n(\mathbb{R})\) consist of the matrices with positive determinant and the matrices with negative determinant.

Problem 2.M.9 Yes, the double cosets partition \(G\). Obviously the double cosets cover \(G\), so we only need to prove that every pair of double cosets is either coincident or disjoint. Let \(g_1, g_2 \in G\) and suppose the double cosets \(Hg_1K, Hg_2K\) are not disjoint, that is, they have some element in common, \(x = h_1 g_1 k_1 = h_2 g_2 k_2\). Then \(g_1 = h_1^{-1} h_2 g_2 k_2 k_1^{-1}\). If \(y\) is some other element of \(Hg_1 K\), that is, \(y = h_3 g_1 k_3\), then \(y = (h_3 h_1^{-1} h_2) g_2 (k_2 k_1^{-1} k_3)\), so \(y \in Hg_2 K\). This establishes \(Hg_1 K \subseteq Hg_2 K\). Analogously we can show that the reverse inclusion also holds. In conclusion, whenever \(Hg_1 K\) and \(Hg_2 K\) are not disjoint, they coincide.

Problem 2.M.10 Suppose first that \(H\) is normal. Then \(Hg = gH\) for each \(g \in G\), therefore \(HgH = gHH = gH\). Conversely, suppose that for all \(g \in G\), \(HgH = gH\). But \(Hg \subseteq HgH = gH\). Also, \(Hg^{-1} \subseteq Hg^{-1}H = g^{-1}H\), and inverting both sides yields \(gH \subseteq Hg\). Since these inclusions hold for all \(g\), the left and right cosets coincide and \(H\) is normal.

Problem 2.M.11

- The matrices \(L\) and \(U\) are computed by using row reduction to bring the matrix \(A\) to upper triangular form. This proceeds one column at a time, starting from the leftmost column. If the entry \(A_{cc}\) vanishes, then give up. Write \(A_{cc} = EB\) where \(E\) is the elementary matrix of the third type with \(E_{cc} = A_{cc}\); then \(B\) will be the same as \(A\) but with all entries in row \(c\) scaled by \(1/c\), therefore \(B_{cc} = 1\). We then use \(B\) in place of \(A\) and proceed to zero out all subdiagonal elements in column \(c\). Namely, if \(A_{rc}\) needs to be zeroed out, write \(A = EB\) where \(E\) is an elementary matrix of the first type with \(E_{rc} = -A_{rc}\); the matrix \(B\) is obtained by subtracting \(A_{rc}\) times row \(c\) from row \(r\). After all subdiagonal entries in column \(c\) have been zeroed out, continue to column \(c + 1\). If this procedure succeeds through the last column, the matrix \(U\) left over will be upper triangular with each diagonal element equal to 1, and the product of all the elementary matrices used will be a lower triangular matrix \(L\) since each individual elementary matrix was lower triangular.
- Suppose \(A = L_1 U_1 = L_2 U_2\) where \(A\) is invertible. This implies that \(L_1, L_2, U_1, U_2\) are invertible. Rearrange the equality to yield \(L_1^{-1} L_2 = U_1 U_2^{-1}\). Since the invertible lower triangular matrices form a subgroup of the invertible matrices, and similarly with the upper triangular matrices where all diagonal entries are 1, we have that the LHS is lower triangular and the RHS is upper triangular with all diagonal entries equal to 1. This implies that both sides equal the identity matrix. Therefore \(L_1 = L_2\) and \(U_1 = U_2\).
This solution is due to Michael Joyce [1]; for an alternative approach, see [2].

The algorithm described in part (a) can be modified so that it always succeeds, producing an LPU decomposition. Namely, start with an empty array of pivot rows, \(R = []\), before processing column 1. The invariant is that just before processing column \(c\), the matrix \(A\) is still invertible, the row list contains \(c - 1\) distinct entries, and for each column \(i < c\), the entry \(A_{i,R_i}\) is equal to 1, and all entries \(A_{ij}\) where \(i \notin \{R_1, R_2, \ldots, R_j\}\) vanish. Clearly this is vacuously true just before processing column 1. To process column \(c\), find the smallest \(r\) such that \(A_{rc} \neq 0\) and \(r \notin R\); this will be the pivot for this column.

Crucially, this step cannot fail. Consider \(A\) where the entries \(\{A_{ic} \mid i \notin R\}\) are all zero. Then row reduction could be used to zero out the entries \(A_{ij}\) where \(j < c, i \in R, i \neq R_j\), yielding some matrix \(A'\) which also has the property that \(A'_{rc} = 0\) for all \(r \notin R\). Then column \(c\) of \(A'\) could be written as the linear combination \(\sum_{i=1}^{c-1} A'_{R_i,c} C_i\) where \(C_i\) denotes column \(A'\) (which has 1 in row \(R_i\) and 0 elsewhere), implying \(A'\) is singular. But then \(A\) would also be singular since \(A\) is a product of \(A'\) and elementary matrices, contradicting the assumption that \(A\) is invertible.

Having found the smallest pivot row index \(r\), set \(R_c = r\) then use an elementary matrix of the third type to set \(A_{rc} = 1\) and elementary matrices of the first type to zero out the entries \(A_{ic}\) where \(i \notin R\) and \(i > r\) by subtracting \(A_{ic}\) times row \(r\) from row \(i\). Note that the entries with \(i \notin R\) and \(i < r\) are zero by assumption that we chose the smallest \(r\) possible. This zeroing doesn't affect any of the first \(c-1\) columns, and it results in column \(c\) having the property that the only nonzero entries are those in rows \(\{R_1, R_2, \ldots, R_c\}\), and \(A_{R_c,c} = 1\), and \(A\) is still invertible. So the operations done on each column preserve the invariant. Also note that when zeroing entry \(A_{ic}\) (which, as noted above, only needs to be done when \(i > r\)), the elementary matrix of the first type used will have its nonzero off-diagonal entry in row \(i\) and column \(r\). But \(i > r\), so this elementary matrix is lower triangular.

After this algorithm has completed for row \(c\), the result is a complete pivot list \(R_{1,\ldots, n}\) that is a permutation of \(1, \ldots, n\), and a factorization \(A = LC\) where:

- \(L\) is a product of elementary matrices of the first and third kinds, but as previously noted, each elementary matrix of the first kind used is lower triangular, and therefore \(L\) is also lower triangular;
- \(C_{i,R_i} = 1\) for each \(i \in \{1, \ldots, n\}\) and \(C_{ij} = 0\) whenever \(i \notin \{R_1, \ldots, R_j\}\).

Consider the elements \(C_{R_c, j}\) for each \(c \in \{1, \ldots, n\}\) and \(j < c\). Since \(R_c \notin \{R_1, \ldots, R_j\}\), this implies that the entry \(C_{R_c, j}\) is zero. That is, all entries to the left of each pivot element in \(C\) vanish. Write \(C\) as the product \(PU\) where row \(r\) of \(U\) equals row \(R_r\) of \(C\), and \(P\) is the permutation matrix that sends row \(r\) to row \(R_r\). Now in every row \(R_r\) of \(C\) the entries in columns \(c < r\) vanish and the entry in column \(r\) is 1, so in row \(r\) of \(U\) the entries in columns \(c < r\) vanish and the entry \(U_{rr}\) is 1. That is, \(U\) is upper triangular with ones along the diagonal. In conclusion, \(A = LPU\) where \(L\) is lower triangular, \(P\) is a permutation matrix, and \(U\) is upper triangular with ones along the diagonal.

There is one double coset for each permutation matrix \(P \in GL_n(\mathbb{R})\). To see this, first observe that part (c) implies that each invertible matrix lies in the double coset of some permutation matrix. Thus, it only remains to show that if two permutation matrices \(P_1, P_2 \in GL_n(\mathbb{R})\) are given, and \(P_1 = LP_2 U\) where \(L\) is lower triangular and \(U\) is upper triangular, then \(P_1 = P_2\). Clearly such \(L, U\) are invertible (all diagonal entries are nonzero). Rearrange the equality to yield \(P_1 U^{-1} P_2^{-1} = L\). \(P_2\) is also a permutation matrix. Left-multiplication by a permutation matrix permutes rows of a matrix, while right-multiplication permutes the columns. Therefore, the entries of \(P_1 U^{-1} P_2^{-1}\) are a permutation of those of \(U^{-1}\). Now all diagonal entries of \(U^{-1}\) are nonzero; another way of saying this is that the set of positions \(\{(i, j) \mid U^{-1}_{ij} \neq 0\}\) is a superset of the corresponding set of positions \(\{(i, j) \mid I_{ij} \neq 0\}\) for the identity matrix. This will remain true after permutation: \(\{(i, j) \mid (P_1 I P_2^{-1})_{ij} \neq 0\} \subseteq \{(i, j) \mid (P_1 U^{-1} P_2^{-1})_{ij} \neq 0\}\). So \(P_1 U^{-1} P_2^{-1}\) being lower triangular implies that \(P_1 I P_2^{-1}\) is lower triangular. But this is another permutation matrix \(P_3 = P_1 P_2^{-1}\) and the only lower triangular permutation matrix is the identity matrix. Therefore \(P_1 = P_2\).

*Note from Brian:*According to folklore, an invertible matrix has an LU decomposition if and only if all its leading principal minors are nonzero. I believe this statement can be generalized to LPU decompositions: let an invertible matrix \(A\) be given; then find the lexicographically smallest permutation \(R\) such that for each \(k = 1, 2, \ldots, n\), the \(k \times k\) matrix given by \(B_{ij} = A_{R(i),R(j)}\) is invertible. The permutation matrix is then given by \(P\) as in (c). This therefore gives a characterization of each double coset. But I am too lazy to write up a proof and check whether it's valid.

Problem 2.M.12 The problem statement says that \(r, s\) are positive, but I assume what was meant is that they must be nonnegative.

- Consider the set of all pairs of integers \(r, s\) such that \(n = ra + sb\); if the pair with the least nonnegative \(r\) also has nonnegative \(s\) then the desired nonnegative solution has been found; otherwise, if \(s\) is negative even for the least nonnegative \(r\) then it will have to be negative for any other nonnegative \(r\) as well, so no nonnegative solution exists. Now suppose there is indeed no nonnegative solution for a given \(n, a, b\). Let \(r, s\) be the pair of integers with \(n = ra + sb\) with the least nonnegative \(r\). Then \(r \leq b - 1\), otherwise write \(r' = r - b, s' = s + a\), whereupon \(n = r'a + s'b\) and \(r' \ge 0\), giving a contradiction. By assumption, \(s \leq -1\). Therefore \(n \leq (b-1)a + (-1)b = ab - a - b\). That is, a nonnegative solution always exists when \(n > ab - a - b\).
- Consider \(n = ab - a - b\). Write \(ab - a - b = ra + sb\) and rearrange this to yield \((s+1)b = (b-r-1)a\). Since \(a, b\) are relatively prime, this is satisfied only when \(a \divides s + 1\) and \(b \divides b - r - 1\). If \(s\) is nonnegative, then \(s \geq a - 1\), and substituting this into \(ab - a - b = ra + sb\) yields \(r \leq -1\). So there is no nonnegative solution for \(ab - a - b\). By (a), this is also the greatest \(n\) for which there is no nonnegative solution.

Problem 2.M.13 A move does not change the GCD of the two numbers in the position, so no \((a, b)\) such that \(\gcd(a, b) > 1\) can be reached starting from \((1, 1)\). Also, obviously, if \(a \le 0\) or \(b \le 0\) can be reached starting from \((1, 1)\). That is, it is necessary that \(a \geq 1, b \geq 1, \gcd(a, b) = 1\). We now claim that this is also sufficient. The proof is by induction; the statement to be proved is that for every positive integer \(n \geq 2\), all pairs of positive integers \(a, b\) with \(a + b = n\) and \(\gcd(a, b) = 1\) are reachable from \((1, 1)\). The details are left as an exercise for the reader.

Problem 2.M.14 This problem is similar to Problem 2.M.13. If \(A \in SL_2(\mathbb{Z})\) then \(A_{11}\) and \(A_{21}\) must be relatively prime and we can use induction, this time on the sum of absolute values since some entries may be negative.

*Lemma:* \(-I = (E' E^{-2})^2\). The proof is trivial.

*Claim:* For every integer \(n \geq 1\), every matrix \(A \in
SL_2(\mathbb{Z})\) with \(|A_{11}| + |A_{21}| = n\) is generated by the
matrices \(E\) and \(E'\).

*Base case:* \(|A_{11}| + |A_{21}|\) cannot be zero. Suppose it is
one. There are four cases. If \(A_{11} = 1\) and \(A_{21} = 0\), then since
\(\det A = 1\), we must have \(A_{22} = 1\). Then \(A = E^{A_{12}}\). If
\(A_{11} = -1\) and \(A_{21} = 0\), then \(A = -IB\); the Lemma tells us that
\(-I \in \langle E, E'\rangle\), and \(B\) falls into the previous case. The
remaining two cases, in which \(A_{11} = 0\) and \(A_{21} = \pm 1\), are
similar.

*Inductive case:* Assume \(A_{11}\) and \(A_{21}\) are relatively
prime and let \(n = |A_{11}| + |A_{21}|\). One of these two magnitudes must
be greater than the other. We can write \(A = MB\) where \(M\) is one of the
four elementary matrices \(E\), \(E^{-1}\), \(E'\), \(E'^{-1}\) and \(B\) is
obtained from \(A\) by, respectively, subtracting the second row from the
first, adding the second row to the first, subtracting the first row from the
second, or adding the first row to the second. We choose the appropriate
\(M\) to reduce the sum of the absolute values of the first column: namely,
the target row is the one that contains the larger of the two entries
\(A_{11}, A_{21}\) in absolute value, and the other row is either subtracted or
added in order to bring that entry closer to zero, depending on whether
the signs of the two entries are like or opposite, respectively.
The resulting matrix \(B\) satisfies \(\det B = 1\) and
\(|B_{11}| + |B_{21}| < n\); by the inductive hypothesis,
\(B \in \langle E, E' \rangle\), and therefore \(A \in \langle E, E'\rangle\)
also.

Problem 2.M.15 This problem is similar to the previous two problems. Denote the two given matrices by \(A\) and \(B\), respectively and the semigroup they generate by \(\langle A, B\rangle\). Obviously every matrix in \(\langle A, B\rangle\) has nonnegative entries and unit determinant; we use an induction argument similar to that in Problem 2.M.14 to show that this condition is also sufficient, although for this problem we are also asked to prove uniqueness.

*Lemma:* For every nonnegative integer \(x\), the matrix
\(A^x = \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}\) can be written
as a product of \(A\) and \(B\) matrices in exactly one way.

*Proof of Lemma:* By induction. If \(x = 0\) then the matrix \(A^x\)
is the identity matrix \(I\); this equals an empty product, but if \(I = AM\)
then \(M\) must be \(A^{-1}\), which has a negative entry; this implies that
no product of \(A\)'s and \(B\)'s with \(A\) as the leftmost factor can equal
\(I\). Since \(B^{-1}\) has a negative entry, we can likewise conclude that
the leftmost factor can't be \(B\) either. So uniqueness holds for \(x = 0\)
(only the empty product works). For the inductive case, if \(A^x = BM\) then
\(M = \begin{pmatrix} 1 & x \\ -1 & 1-x\end{pmatrix}\), which has at
least one negative entry, so any product with \(B\) as the leftmost factor is
excluded. And any product with \(A\) as the leftmost factor would have
\(A^{x-1}\) as the product of the remaining factors; the uniqueness of such a
product representation for \(A^{x-1}\), which is the inductive hypothesis,
then implies uniqueness for \(A^x\) as well.

*Claim:* For every positive integer \(n\), every integer matrix
\(C\) with four nonnegative entries, unit determinant, and \(C_{11} + C_{21}
= n\) can be written as a product of \(A\) and \(B\) matrices in exactly one
way.

*Base case:* Suppose \(n = 1\). If \(C_{11} = 1\) and \(C_{21} = 0\),
then the assumption \(\det C = 1\) implies \(C_{22} = 1\), that is,
\(C = \begin{pmatrix} 1 & x \\ 0 & 1\end{pmatrix}\) for some
nonnegative integer \(x\). Then \(C = A^x\). The Lemma states that \(C\) can
only be written as a product of \(A\) and \(B\) matrices in one way (namely
\(A^x\)). On the other hand, the case \(C_{11} = 0\) and \(C_{21} = 1\) does
not occur (such a matrix would have to have \(C_{12} < 0\) in order to
have unit determinant).

*Inductive case:* If \(C_{11} > C_{21}\) then the assumption
\(\det C > 0\) implies that \(C_{12} \ge C_{22}\). Write \(C =
AD\); the matrix \(D\) is obtained from \(C\) by subtracting the second row
from the first, and since \(C_{12} \ge C_{22}\), it is also the case that
\(D\) has unit determinant and four nonnegative entries. The sum of the
entries in the first column of \(D\) will be less than \(n\), so by the
inductive hypothesis, there is exactly one way to write \(C\) as a product of
\(A\)'s and \(B\)'s with \(A\) as the leftmost factor. On the other hand, there
are no ways with \(B\) as the leftmost factor, since if \(C = BD\) then the
matrix \(D\) would have at least one negative entry. Uniqueness is therefore
established for such \(C\). If on the other hand \(C_{21} > C_{11}\), then
we would analogously get one product with \(B\) as the leftmost factor, where
the remaining part equals the matrix obtained by subtracting the first row from
the second, and no representations with \(A\) as the leftmost factor.
Finally, if \(C_{11} = C_{21}\), which is only
possible when both are equal to unity, the unique choice of the first factor in
the representation depends on which of \(C_{12}\) and \(C_{22}\) is greater
(they can't be equal if the determinant is 1).