3.1 Invertibility
Reading
Try out the Preview Activity and read Chapter 3 Section 1 in Understanding Linear Algebra by David Austin.
An nxn matrix A is invertible if there is a matrix B so that [latex]BA=I_n[/latex].
An nxn matrix A is invertible if and only if the reduced row echelon form of A is [latex]I_n[/latex].
Important Properties of Invertible Matrices
- If A is invertible, then [latex]A\vec{x}=\vec{b}[/latex] has the unique solution [latex]\vec{x} = A^{-1}\vec{b}[/latex]
- If A is 2x2 there is an easy formula for it's inverse, that should be memorized: If [latex]A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}[/latex], then [latex]A^{-1}= \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}[/latex]
- If A is an nxn invertible matrix, then we can find the inverse using row reduction: [latex][A|I_n] \rightarrow \dots \rightarrow [I|A^{-1}][/latex].
- If A, B are nxn invertible matrices, then AB is invertible and [latex](AB)^{-1} = B^{-1}A^{-1}[/latex] (Compare this with finding the inverse of a composition: If f and g are invertible functions and [latex]f\circ g[/latex] is defined, then it is invertible, and [latex](f\circ g)^{-1} = g^{-1}\circ f^{-1}[/latex].
Computation Examples:
Finding 2x2 matrix inverse, from formula
Finding 2x2 matrix inverse, by row reduction
Finding 3x3 matrix inverse, by row reduction, using Sage
Finding 3x3 matrix inverse, by row reduction, showing row reduction
