Linear Algebra

Section IV. Matrix Operations 233we do Gauss-Jordan reduction, meanwhile performing the same operations onthe identity. For clerical convenience we write the matrix and the identityside-by-side, and do the reduction steps together.()()1 1 1 0 −2ρ 1 +ρ 2 1 1 1 0−→2 −1 0 10 −3 −2 1()−1/3ρ 2 1 1 1 0−→0 1 2/3 −1/3()−ρ 2 +ρ 1 1 0 1/3 1/3−→0 1 2/3 −1/3This calculation has found the inverse.( ) −1 ( )1 1 1/3 1/3=2 −1 2/3 −1/34.9 Example This one happens to start with a row swap.⎛⎞⎛⎞0 3 −1 1 0 01 0 1 0 1 0⎜⎟⎝1 0 1 0 1 0⎠ ρ 1↔ρ 2 ⎜⎟−→ ⎝0 3 −1 1 0 0⎠1 −1 0 0 0 11 −1 0 0 0 1⎛−ρ 1 +ρ 3 ⎜1 0 1 0 1 0⎞⎟−→ ⎝0 3 −1 1 0 0⎠0 −1 −1 0 −1 1.−→⎛⎜1 0 0 1/4 1/4 3/4⎞⎟⎝0 1 0 1/4 1/4 −1/4⎠0 0 1 −1/4 3/4 −3/44.10 Example We can detect a non-invertible matrix when the left half won’treduce to the identity.() ()1 1 1 0 −2ρ 1 +ρ 2 1 1 1 0−→2 2 0 10 0 −2 1With this procedure we can give a formula for the inverse of a general 2×2matrix, which is worth memorizing. But larger matrices have more complexformulas so we will wait for more explanation in the next chapter.4.11 Corollary The inverse for a 2×2 matrix exists and equals(acif and only if ad − bc ≠ 0.) −1 (b 1 d=d ad − bc −c)−ba