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 2×2 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 2×2 matrix inverse, from formula
Finding 2×2 matrix inverse, by row reduction
Finding 3×3 matrix inverse, by row reduction, using Sage
Finding 3×3 matrix inverse, by row reduction, showing row reduction