Linear Algebra

Section III. Other Formulas 3231.10 Example Ifthen the adjoint adj(T ) is⎛⎛⎝ T ⎞∣ 1 −10 1 ∣1,1 T 2,1 T 3,1T 1,2 T 2,2 T 3,2⎠=−∣ 2 −11 1 ∣T 1,3 T 2,3 T 3,3 ⎜⎝∣ 2 11 0∣⎛1 0⎞4T = ⎝2 1 −1⎠1 0 1−∣ 0 40 1∣∣ 1 41 1∣−∣ 1 01 0∣∣ 0 4⎞1 −1∣−∣ 1 4⎛2 −1∣= ⎝ 1 0 −4⎞−3 −3 9 ⎠∣ 1 0⎟ −1 0 1⎠2 1∣and taking the product with T gives the diagonal matrix |T | · I.⎛⎝ 1 0 4 ⎞ ⎛2 1 −1⎠⎝ 1 0 −4 ⎞ ⎛−3 −3 9 ⎠ = ⎝ −3 0 0 ⎞0 −3 0 ⎠1 0 1 −1 0 1 0 0 −31.11 Corollary If |T | ̸= 0 then T −1 = (1/|T |) · adj(T ).1.12 Example The inverse of the matrix from Example 1.10 is (1/−3)·adj(T ).⎛⎞ ⎛⎞1/−3 0/−3 −4/−3 −1/3 0 4/3T −1 = ⎝−3/−3 −3/−3 9/−3 ⎠ = ⎝ 1 1 −3 ⎠−1/−3 0/−3 1/−3 1/3 0 −1/3The formulas from this section are often used for by-hand calculation andare sometimes useful with special types of matrices. However, they are not thebest choice for computation with arbitrary matrices because they require morearithmetic than, for instance, the Gauss-Jordan method.Exerciseš 1.13 Find the cofactor.T =( 1 0) 2−1 1 30 2 −1(a) T 2,3 (b) T 3,2 (c) T 1,3̌ 1.14 Find the determinant by expanding3 0 11 2 2∣−1 3 0∣(a) on the first row (b) on the second row (c) on the third column.1.15 Find the adjoint of the matrix in Example 1.6.̌ 1.16 Find the matrix adjoint to each.