P =
(
cos α sin α
−sin α cos α
)
and
P + PT = I,
then the value of α, where 0 ≤ α ≤ π/2, is ______.
(A) π/2
(B) π/3
(C) 3π/2
(D) 0
Find α If P + PT = I for the Given Trigonometric Matrix
Understanding the Given Matrix Equation
This question combines two important mathematical concepts: the transpose of a matrix and basic trigonometric values. We are given a 2 × 2 matrix P containing cos α and sin α, together with the condition that the sum of P and its transpose PT is equal to the identity matrix I.
The given matrix is:
P =
(
cos α sin α
−sin α cos α
)
We need to calculate PT, add it to P, compare the resulting matrix with the identity matrix, and then determine the value of α within the specified interval:
0 ≤ α ≤ π/2
The key feature of the problem is that the off-diagonal terms cancel when P and PT are added. This reduces the matrix equation to the simple trigonometric equation 2 cos α = 1.
What Is the Transpose of a Matrix?
The transpose of a matrix is obtained by interchanging its rows and columns. If an element appears in the first row and second column of the original matrix, it moves to the second row and first column in the transpose.
For a general 2 × 2 matrix:
A =
(
a b
c d
)
the transpose is:
AT =
(
a c
b d
)
Therefore, taking a transpose does not change the diagonal elements, but the two off-diagonal elements exchange their positions.
Step-by-Step Solution
Step 1: Write the Given Matrix P
The given matrix is:
P =
(
cos α sin α
−sin α cos α
)
The first row contains cos α and sin α, while the second row contains −sin α and cos α.
We are given the condition:
P + PT = I
Therefore, the first required step is to calculate the transpose of P.
Step 2: Find the Transpose PT
To obtain the transpose, we interchange the rows and columns of P. Thus:
PT =
(
cos α −sin α
sin α cos α
)
Notice that the diagonal elements cos α remain in their original positions. The element sin α moves from the first row, second column to the second row, first column. Similarly, −sin α moves from the second row, first column to the first row, second column.
Step 3: Calculate P + PT
Now add the corresponding elements of P and PT:
P + PT
=
(
cos α sin α
−sin α cos α
)
+
(
cos α −sin α
sin α cos α
)
Adding the elements in the first row and first column gives:
cos α + cos α = 2 cos α
Adding the elements in the first row and second column gives:
sin α + (−sin α) = 0
Similarly, the second row and first column gives:
−sin α + sin α = 0
Finally, the second row and second column gives:
cos α + cos α = 2 cos α
Therefore:
P + PT
=
(
2 cos α 0
0 2 cos α
)
Step 4: Compare the Result with the Identity Matrix
The identity matrix of order 2 is:
I =
(
1 0
0 1
)
According to the given condition:
P + PT = I
Therefore:
(
2 cos α 0
0 2 cos α
)
=
(
1 0
0 1
)
Two matrices are equal only when their corresponding elements are equal. Comparing the diagonal elements gives:
2 cos α = 1
Therefore:
cos α = 1/2
Step 5: Find the Value of α
We now need to determine the angle whose cosine is 1/2. From standard trigonometric values:
cos(π/3) = 1/2
Therefore:
α = π/3
The question specifies the interval:
0 ≤ α ≤ π/2
The angle π/3 lies within this interval because:
0 ≤ π/3 ≤ π/2
Hence, the required value is:
α = π/3
Complete Matrix Calculation in Compact Form
The complete calculation can be summarized as:
P =
(
cos α sin α
−sin α cos α
)
Therefore:
PT =
(
cos α −sin α
sin α cos α
)
Adding the two matrices:
P + PT
=
(
2 cos α 0
0 2 cos α
)
Since P + PT = I:
2 cos α = 1
Therefore:
cos α = 1/2
Hence:
α = π/3
Why the Sine Terms Disappear
The off-diagonal elements of the original matrix are sin α and −sin α. When the matrix is transposed, these elements exchange their positions. As a result, every off-diagonal position contains the sum of sin α and −sin α.
Therefore:
sin α − sin α = 0
This is why the resulting matrix contains zeros in both off-diagonal positions. Only the diagonal terms remain, and each diagonal term becomes 2 cos α.
Thus, the entire matrix condition reduces to the single scalar equation:
2 cos α = 1
Detailed Analysis of Each Option
Option (A): π/2
This option is incorrect because:
cos(π/2) = 0
Therefore:
2 cos(π/2) = 0
This would make P + PT equal to the zero matrix rather than the identity matrix. Hence, α = π/2 does not satisfy the given condition.
Option (B): π/3
This option is correct because:
cos(π/3) = 1/2
Therefore:
2 cos(π/3) = 2 × 1/2 = 1
Hence:
P + PT
=
(
1 0
0 1
)
= I
Thus, α = π/3 satisfies both the matrix equation and the given interval.
Option (C): 3π/2
This option is incorrect for two reasons. First, 3π/2 does not lie in the specified interval 0 ≤ α ≤ π/2. Second:
cos(3π/2) = 0
Therefore, it cannot satisfy the required equation 2 cos α = 1.
Option (D): 0
This option is incorrect because:
cos 0 = 1
Therefore:
2 cos 0 = 2
This would give:
P + PT = 2I
rather than I. Hence, α = 0 does not satisfy the given matrix equation.
Verification of the Correct Answer
For α = π/3:
cos(π/3) = 1/2
and:
sin(π/3) = √3/2
Therefore:
P =
(
1/2 √3/2
−√3/2 1/2
)
Its transpose is:
PT =
(
1/2 −√3/2
√3/2 1/2
)
Adding the matrices gives:
P + PT
=
(
1 0
0 1
)
Therefore:
P + PT = I
This directly verifies that α = π/3 is the correct answer.
Final Answer
The value of α satisfying P + PT = I in the interval 0 ≤ α ≤ π/2 is π/3.
Correct Option: (B) π/3
