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2×2 invertible upper triangular matrices form a subgroup of general linear group


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.10

Let $G = \left\{ \left[{a \atop 0} {b \atop c}\right] \ |\ a,b,c \in \mathbb{R}, a \neq 0, c \neq 0 \right\}$.

(1) Show that $G$ is closed under matrix multiplication.
(2) Find the inverse of an arbitrary $G$ element and show that $G$ is closed under inverses.
(3) Deduce that $G$ is a subgroup of $GL_2(\mathbb{R})$.
(4) Prove that the set of all matrices in $G$ with $a = c$ is also a subgroup of $GL_2(\mathbb{R})$.


Solution:

(1) Let $A = \left[{a_1 \atop 0} {b_1 \atop c_1}\right]$, $B = \left[{a_2 \atop 0} {b_2 \atop c_2}\right] \in G$. Then $$AB = \left[{a_1a_2 \atop 0} {a_1b_2 + b_1c_2 \atop c_1c_2}\right],$$ which is also in $G$. So $G$ is closed under matrix multiplication.

(2) Let $A = \left[{a \atop 0} {b \atop c}\right] \in G$. Note that $$B = \left[{a^{-1} \atop 0} {\frac{-b}{ac} \atop c^{-1}}\right] \in G,$$ and that $AB = I$. Thus $G$ is closed under inversion.

(3) Since the identity matrix is in $G$, $G$ is nonempty. Thus $G$ is a subgroup of $GL_2(\mathbb{R})$.

(4) Consider the subset $H \subseteq G$ consisting of those matrices whose diagonal entries are equal. We can see by the above calculations that $H$ is also closed under matrix multiplication and inversion, so that $H$ is a subgroup of both $G$ and $GL_2(\mathbb{R})$.


Linearity

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