Answer to Question #190376 in Linear Algebra for Regomoditswe Dibob

Question

Consider the given matrix B =   2 2 0 1 0 1 0 1 1   . Find detB and use it to determine whether or not B is invertible, and if so, find B −1 . (Hint: Use the matrix equation BX = I)

Answer 1

            | 2 2 0 |
det B = det | 1 0 1 | =
            | 0 1 1 |

2*0*1 + 2*1*0 + 0*1*1 - 0*0*0 - 2*1*1 - 2*1*1 = -4
As the determinant is not zero, the matrix B is invertable and

          1      | C_11 C_12 C_13 |T
B^(-1) = ----- * | C_21 C_22 C_23 |    , wher C is a matrix of cofactors
         det B   | C_31 C_32 C_33 |

C_11 = det|0 1| = -1    C_12 = -det|1 1| = -1  C_13 = det|1 0| = 1
          |1 1]                    |0 1|                 |0 1|

C_21 = -det|2 0| = -2    C_22 = det|2 0| = 2  C_23 = -det|2 2| = -2
           |1 1]                   |0 1|                 |0 1|         

C_31 = det|2 0| = 2    C_32 = -det|2 0| = -2  C_33 = det|2 2| = -2
          |0 1]                   |1 1|                 |1 0|          

              1 |-1 -2  2|
So B^(-1)  = ---|-1  2 -2|
             -4 | 1 -2 -2|