What is \(GL_n(\mathbb R)\)?

What is the identity element in a matrix group under multiplication?

What determinant condition tells you a square matrix is in \(GL_n(\mathbb R)\)?

Is a \(2\times3\) real matrix an element of \(GL_2(\mathbb R)\)?

Determine whether \(\begin{pmatrix}1&0\\0&2\end{pmatrix}\) is in \(GL_2(\mathbb R)\).

Determine whether \(\begin{pmatrix}1&2\\2&4\end{pmatrix}\) is in \(GL_2(\mathbb R)\).

For \(A=\begin{pmatrix}2&0\\0&3\end{pmatrix}\), find \(A^{-1}\).

Why is matrix multiplication generally not commutative?

Compute the determinant of the rotation matrix \(R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}\).

Show that the product of two invertible matrices is invertible by giving its inverse.

Verify that \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) is in \(GL_2(\mathbb R)\) and interpret it geometrically.

Give an example of two invertible \(2\times2\) matrices \(A,B\) with \(AB\ne BA\).

Verify the group axioms for \(GL_n(\mathbb R)\) under multiplication.

Show that the set of \(2\times2\) real matrices with determinant \(1\) is closed under multiplication.

Show that a \(2\times2\) matrix with determinant \(1\) has an inverse that also has determinant \(1\).

Explain why all real \(2\times2\) matrices do not form a group under multiplication.

For \(A=\begin{pmatrix}1&2\\0&1\end{pmatrix}\), compute \(A^n\) for positive integers \(n\).

A student says \(A^{-1}\) is found by taking the reciprocal of every entry of \(A\). Diagnose the error.

Prove that if \(A\in GL_n(\mathbb R)\), then \(A^{-1}\in GL_n(\mathbb R)\).

Explain why rotation matrices form a subgroup of \(GL_2(\mathbb R)\).