Question below.?

Question

given the matrix X

1 x x x
x 1 x x
x x 1 x
x x x 1 find x* element of R such that X is singular
find x** element of R such that X is nonsingular



Given matirix T

sint -cost -cost
cost sint -sint
0 0 1

for some t element of R

find if the inverse of T (if it exist)

Answer 1

Well, I'll answer all but the inverse of T. I need to
work on that and I'll come back here.
Question 1: If x* = 1 then X is singular, because
all 4 rows are the same. That makes det[X] = 0
so X is singular.
If x* = 0, then X is the identity matrix, so it is
nonsingular

Question 2. The inverse of T does exist.
Expand the determinant by minors of
the 3rd row. Det[T] = 1(sin²t + cos²t) = 1.
Probably the easiest way to get the inverse
is to use T^-1 = T(adjoint)/det[T] = T(adjoint)
Now I'll check it out, but I think element a_(ij)
of the adjoint is computed by taking
the signed minor of element a_ij of T and
then transposing the resulting matrix
Hope that helps!

Answer 2

given the matrix X

1 x x x
x 1 x x
x x 1 x
x x x 1 find x* element of R such that X is singular
find x** element of R such that X is nonsingular

if x=1, then the matrix is singular
if x=0 then thematrix is the identity and non-singular .

Given matirix T

sint -cost -cost
cost sint -sint
0 0 1

for some t element of R

find if the inverse of T (if it exist)

the matrix T has an inverse because its determinant is = 1, i.e. not equal to zero. .