LetA=cosθ0sinθ010−sinθ0cosθ.
If for some
θ∈(0,π),A2=A
T, then the sum of the diagonal elements of the matrix
(A+I)3+(A−I)3−6Aisequal to:
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Step-by-Step Solution
Step 1: Calculate
A2
and
AT
First, we need to calculate the square of matrix
A
, denoted as
A2
, by multiplying
A
by itself. We also need to find the transpose of matrix
A
, denoted as
AT
, by swapping its rows and columns.
Step 2: Equate
A2
and
AT
to find
θ
Given that
A2=AT
, we equate the corresponding elements of the two matrices. This gives us a system of trigonometric equations. Solving these equations for
θ
within the given interval
(0,π)
yields
θ=32π
.
Step 3: Simplify the expression
(A+I)3+(A−I)3−6A
We expand the cubic terms
(A+I)3
and
(A−I)3
using the binomial expansion formula
(x+y)3=x3+3x2y+3xy2+y3
and
(x−y)3=x3−3x2y+3xy2−y3
. Since
I
is the identity matrix,
AI=A
and
In=I
. After expansion, many terms cancel out, simplifying the expression to
2A3
.
Step 4: Calculate
A3
Since we found that
A2=AT
, we can substitute
AT
for
A2
in the expression for
A3
. Thus,
A3=A2⋅A=AT⋅A
. Performing this matrix multiplication, we find that
ATA
results in the identity matrix
I
.
Step 5: Calculate the final expression and sum of diagonal elements
Substituting
A3=I
into the simplified expression
2A3
, we get
2I
. The identity matrix
I
has ones on its diagonal, so
2I
has twos on its diagonal. The sum of the diagonal elements of
2I
is
2+2+2=6
.