Linear Algebra

322 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 34 6∣ = 12 − 60 + 48 = 01.7 Example A row or column with many zeroes suggests a Laplace expansion.1 5 0∣ ∣ ∣ ∣∣∣ 2 1 1∣3 −1 0∣ = 0 · (+1) 2 1∣∣∣ 3 −1∣ + 1 · (−1) 1 5∣∣∣ 3 −1∣ + 0 · (+1) 1 52 1∣ = 16We finish by applying this result to derive a new formula for the inverseof a matrix. With Theorem 1.5, the determinant of an n × n matrix T canbe calculated by taking linear combinations of entries from a row and theirassociated cofactors.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 the “wrong” row — row kwith 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 equation summarizes (∗) and (∗∗).⎛⎞ ⎛⎞ ⎛⎞t 1,1 t 1,2 . . . t 1,n T 1,1 T 2,1 . . . T n,1 |T | 0 . . . 0t 2,1 t 2,2 . . . t 2,nT 1,2 T 2,2 . . . T n,2⎜⎝⎟ ⎜.⎠ ⎝⎟.⎠ = 0 |T | . . . 0⎜⎝⎟. ⎠t n,1 t n,2 . . . t n,n 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