Use the properties of determinants to prove that if Ak 0 th

Use the properties of determinants to prove that if A^k = 0, then det(A) = 0. Are nilpotent matrices invertible? Justify your answer. If A is nilpotent of index 2, prove that the inverse of I - A is I + A.

Solution

7. For any two matrices A and B , the det( A X B ) = det(A) X det (B)

Thus det ( A^k ) = det (A) X det (A) .........................(k times)...........X det (A)

if A^k = 0

Then det (A^k) = 0

Thus det (A) = 0

8. Nilpotent matrix is a square matrix of order k such that A^k = 0

We proved in the previous question that if A^k = 0 then det (A) = 0

Since if the determinant of any Matrix is zero, then it is not invertible. Hence nilpotent matrices are non invertable.

9. Nilpotent matrix of order : A² = 0

( I - A ) X ( I + A ) = I² - A X I + I X A - A²

= I - A + A - A²

= I

Therefore inverse of ( I - A ) is ( I + A )