Calculate the following determinants using cofactors. a. det 1 3 5 6 4 2 2 5 1 b. det 1 1 0 1 0 2 1 1 2 2 2 3 0 0 6 2 c. det 1 4 1 3 2 10 0 1 0 0 2 2 0 0 2 1 d. det 2 1 0 0 0 1 2 1 0 0 0 1 2 1 0 0 0 1 2 1 0 0 0 1 2

S343 Section 3.4 Notes- Repeated Roots of the Characteristic Equation; Reduction of Order 10-4-16 For + + = 0 and characteristic equation + + = 0, repeated roots occur when − 4 = 0 − o Follows from quadratic formula that 1 = 2= 2 − o Both roots give solution1 = 2, not obvious how to find second solution ′′ ′ Ex. 2 − 6 + 9 = 0 o − 6 + 9 = 0 ( − 3 − 3 = 0 = 3 with multiplicity 2 3 o