True or false

Question

 If A is a square matrix with inverse A^-1 and c is a nonzero real number, then  ( cA)^-1=1/c A^-1
 

 

If you think the statement is true, then show that it is true. On the other hand, if you think the statement is false, then give an example that disproves the statement.

Answer 1

Yes, it is true. Think of cA as the product cI_n•A. Then we have

(cA)^-1=(cI_n•A)^-1=A^-1•(cI_n)^-1=(cI_n)^-1A^-1=(c^-1)I_nA^-1=c^-1A^-1.