Linear Algebra

Section IV. Matrix Operations 225We have seen how to produce a matrix that will swap rows. Multiplying by thispermutation matrix swaps the first and third rows.⎛⎜0 0 1⎞ ⎛⎞ ⎛⎞0 2 1 1 1 0 2 0⎟ ⎜⎟ ⎜⎟⎝0 1 0⎠⎝0 1 3 −3⎠ = ⎝0 1 3 −3⎠1 0 0 1 0 2 0 0 2 1 1To see how to perform a row combination, we observe something about thosetwo examples. The matrix that rescales the second row by a factor of threearises in this way from the identity.⎛ ⎞ ⎛1 0 0⎜ ⎟⎝0 1 0⎠ 3ρ 2 ⎜1 0 0⎞⎟−→ ⎝0 3 0⎠0 0 1 0 0 1Similarly, the matrix that swaps first and third rows arises in this way.⎛⎜1 0 0⎞ ⎛⎟⎝0 1 0⎠ ρ 1↔ρ 3 ⎜0 0 1⎞⎟−→ ⎝0 1 0⎠0 0 1 1 0 03.18 Example The 3×3 matrix that arises as⎛⎜1 0 0⎞ ⎛⎟⎝0 1 0⎠ −2ρ 2+ρ 3 ⎜1 0 0⎞⎟−→ ⎝0 1 0⎠0 0 10 −2 1will, when it acts from the left, perform the combination operation −2ρ 2 + ρ 3 .⎛ ⎞ ⎛⎞ ⎛⎞1 0 0 1 0 2 0 1 0 2 0⎜ ⎟ ⎜⎟ ⎜⎟⎝0 1 0⎠⎝0 1 3 −3⎠ = ⎝0 1 3 −3⎠0 −2 1 0 2 1 1 0 0 −5 73.19 Definition The elementary reduction matrices result from applying a oneGaussian operation to an identity matrix.(1) I kρ i−→ M i (k) for k ≠ 0(2) I ρ i↔ρ j−→ Pi,j for i ≠ j(3) I kρ i+ρ j−→Ci,j (k) for i ≠ j3.20 Lemma Gaussian reduction can be done through matrix multiplication.(1) If H kρ i−→ G then M i (k)H = G.(2) If H ρ i↔ρ j−→ G then Pi,j H = G.(3) If H kρ i+ρ j−→ G then Ci,j (k)H = G.