Linear Algebra : Zero Matrix Proof
Not what you're looking for? Search our solutions OR ask your own Custom question.
Let A be a 2 x 2 matrix with A^3 = O. Prove that A^2 = O. Where O is the zero matrix. (Note: I've said nothing about the invertibility of A, it may or may not be invertible).
© BrainMass Inc. brainmass.com March 6, 2023, 1:13 pm ad1c9bdddfhttps://brainmass.com/math/linear-algebra/linear-algebra-zero-matrix-proof-8608
Solution Preview
*If A is invertible, i.e. |A|<>0
then we have: (A)^-1 * A*A*A = (A)^-1 * 0
so A*A=0
*If A isn't invertible, so |A|=0
i.e. a11*a22 - a12*a21=0
or a11*a22 = a12*a21
then A*A=
( a11^2+a12*a21 a11*a12+a12*a22
a11*a21 +a21*a22 a22^2*a12*a21)
And let A*A*A=
(B11 ...
Solution Summary
A proof is offered for an equality to the zero matrix. The invertibility of a matrix function is given.
$2.49