Show that no 3x3 matrix A exists such that A^2 + I = 0
Find a 2x2 matrix A with this property. Please explain how to go about this?
2007-02-12 16:32:52 · 4 answers · asked by Red Ruby 1 in Science & Mathematics ➔ Mathematics
first: A=
0 -1
1. 0
has the property that A^2 + I =0.
if A is 3x3, then to have A^2 + I =0 means that A^2 = -I
but then
det (A^2 ) = det (-I)
det( A^2) =det(A)^2
while det( -I) = (-1)^3 = -1
so we get det(A)^2 = -1
but this has no solutions, since det(A) ^2 >=0, over the real numbers.
2007-02-16 11:05:11 · answer #1 · answered by Anonymous · 3⤊ 2⤋
actually, there are infinite 3x3 matrices which verifies A^2+1=0
the simplest is
(i, 0 0;
0, i, 0;
0,0,i)
where "i" is the imaginary constant
for 2x2 matrices the problem is easier:
besides the usual solution (i,0;0,i)
the matrices you are looking for have the form
(-i, 0;
x, +i)
or
(+i, 0;
x, -i)
where "x" is an arbitrary constant..
to work this out.. a bit of fantasy and a lot of algebra!
2007-02-13 01:06:05 · answer #2 · answered by Thor2007 2 · 1⤊ 1⤋
Take the identity matrix with the imaginary i instead of 1.
2007-02-13 00:50:06 · answer #3 · answered by bruinfan 7 · 2⤊ 0⤋
Fundamentaly it is
[A]^2 = [I]^-1
for all sqare matrix
2007-02-13 00:44:15 · answer #4 · answered by Mritunjay 2 · 2⤊ 0⤋