If A is an invertible matrix, then
is invertible, and

.

Question

If A is an invertible matrix, then 
  is invertible, and 
  
.

Quizplus · Accepted Answer

If A is invertible, then it has an inverse matrix A^-1. We can use matrix algebra to show that (11edaba5_20aa_2bda_a975_7b5d1d076c5a_TBMC1039_00)(A^-1)(A)(11edaba5_20aa_2bdb_a975_e569f4f95eae_TBMC1039_00) = (11edaba5_20aa_2bda_a975_7b5d1d076c5a_TBMC1039_00)(I)(11edaba5_20aa_2bdb_a975_e569f4f95eae_TBMC1039_00) = 11edaba5_20aa_2bda_a975_7b5d1d076c5a_TBMC1039_00(11edaba5_20aa_2bdb_a975_e569f4f95eae_TBMC1039_00), which equals the identity matrix I. Therefore, (11edaba5_20aa_2bda_a975_7b5d1d076c5a_TBMC1039_00)(A^-1)(A) is the inverse of 11edaba5_20aa_2bdb_a975_e569f4f95eae_TBMC1039_00, which means that 11edaba5_20aa_2bdb_a975_e569f4f95eae_TBMC1039_00 is invertible.

If a Is an Invertible Matrix, Then