3.1	Invertibility

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

B A = I_{n}

.

An nxn matrix A is invertible if and only if the reduced row echelon form of A is

I_{n}

.

Important Properties of Invertible Matrices

If A is invertible, then $A \vec{x} = \vec{b}$ has the unique solution $\vec{x} = A^{- 1} \vec{b}$
If A is 2×2 there is an easy formula for it’s inverse, that should be memorized: If $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , then $A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$
If A is an nxn invertible matrix, then we can find the inverse using row reduction: $[A | I_{n}] \to \dots \to [I | A^{- 1}]$ .
If A, B are nxn invertible matrices, then AB is invertible and $(A B)^{- 1} = B^{- 1} A^{- 1}$ (Compare this with finding the inverse of a composition: If f and g are invertible functions and $f \circ g$ is defined, then it is invertible, and $(f \circ g)^{- 1} = g^{- 1} \circ f^{- 1}$ .

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

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