Linear Algebra

328 Chapter Four. DeterminantsAlternatively, we can expand down the second column.∣ ∣ |T| = 2 · (−1)4 6∣∣∣∣ ∣7 9∣ + 5 · (+1) 1 3∣∣∣∣ 7 9∣ + 8 · (−1) 1 3= 12 − 60 + 48 = 04 6∣ 1.7 Example A row or column with many zeroes suggests a Laplace expansion.1 5 0∣ ∣ 2 1∣∣∣∣ 2 1 1= 0 · (+1)∣3 −1∣ + 1 · (−1) 1 5∣∣∣∣ 3 −1∣ + 0 · (+1) 1 52 1∣ = 16∣3 −1 0∣We finish by applying this result to derive a new formula for the inverseof a matrix. With Theorem 1.5, we can calculate the determinant of an n×nmatrix T by taking linear combinations of entries from a row and their associatedcofactors.t i,1 · T i,1 + t i,2 · T i,2 + · · · + t i,n · T i,n = |T|(∗)Recall that a matrix with two identical rows has a zero determinant. Thus, forany matrix T, weighing the cofactors by entries from row k with k ≠ i gives zerot i,1 · T k,1 + t i,2 · T k,2 + · · · + t i,n · T k,n = 0(∗∗)because it represents the expansion along the row k of a matrix with row i equalto row k. This summarizes (∗) and (∗∗).⎛⎜⎝⎞t 1,1 t 1,2 . . . t 1,nt 2,1 t 2,2 . . . t 2,n⎟.⎠t n,1 t n,2 . . . t n,n⎛⎞ ⎛⎞T 1,1 T 2,1 . . . T n,1 |T| 0 . . . 0T 1,2 T 2,2 . . . T n,2⎜⎟⎝ .⎠ = 0 |T| . . . 0⎜⎟⎝ . ⎠T 1,n T 2,n . . . T n,n 0 0 . . . |T|Note that the order of the subscripts in the matrix of cofactors is opposite tothe order of subscripts in the other matrix; e.g., along the first row of the matrixof cofactors the subscripts are 1, 1 then 2, 1, etc.1.8 Definition The matrix adjoint to the square matrix T is⎛⎞T 1,1 T 2,1 . . . T n,1T 1,2 T 2,2 . . . T n,2adj(T) =⎜⎟⎝ .⎠T 1,n T 2,n . . . T n,nwhere T j,i is the j, i cofactor.1.9 Theorem Where T is a square matrix, T · adj(T) = adj(T) · T = |T| · I.Proof Equations (∗) and (∗∗).QED