If both $A - \dfrac{I}{2}$ and $A + \dfrac{I}{2}$ are orthogonal matrix, then which of the following statements are <b>incorrect</b>? (where $I$ is an identity matrix order same as that of $A$.)
Step-by-Step Solution
Key Concept: The key idea is to use the definition of an orthogonal matrix ($P^T P = I$) for both given matrices, then combine the resulting matrix equations through addition and subtraction to deduce fundamental properties of matrix A.
Step 1: Let $P = A - \frac{I}{2}$ and $Q = A + \frac{I}{2}$. Since $P$ and $Q$ are orthogonal matrices, we have $P^T P = I$ and $Q^T Q = I$.
From $P^T P = I$:
$$ \left(A - \frac{I}{2}\right)^T \left(A - \frac{I}{2}\right) = I $$
$$ \left(A^T - \frac{I}{2}\right) \left(A - \frac{I}{2}\right) = I $$
$$ A^T A - \frac{1}{2}A^T - \frac{1}{2}A + \frac{1}{4}I = I \quad \cdots (1) $$
From $Q^T Q = I$:
$$ \left(A + \frac{I}{2}\right)^T \left(A + \frac{I}{2}\right) = I $$
$$ \left(A^T + \frac{I}{2}\right) \left(A + \frac{I}{2}\right) = I $$
$$ A^T A + \frac{1}{2}A^T + \frac{1}{2}A + \frac{1}{4}I = I \quad \cdots (2) $$
Step 2: Add equations (1) and (2):
$$ \left(A^T A - \frac{1}{2}A^T - \frac{1}{2}A + \frac{1}{4}I\right) + \left(A^T A + \frac{1}{2}A^T + \frac{1}{2}A + \frac{1}{4}I\right) = I + I $$
$$ 2A^T A + \frac{1}{2}I = 2I $$
$$ 2A^T A = \frac{3}{2}I $$
$$ A^T A = \frac{3}{4}I $$
Step 3: Subtract equation (1) from equation (2):
$$ \left(A^T A + \frac{1}{2}A^T + \frac{1}{2}A + \frac{1}{4}I\right) - \left(A^T A - \frac{1}{2}A^T - \frac{1}{2}A + \frac{1}{4}I\right) = I - I $$
$$ A^T + A = 0 $$
$$ A^T = -A $$
Thus, $A$ is a skew-symmetric matrix.
Step 4: Substitute $A^T = -A$ into $A^T A = \frac{3}{4}I$:
$$ (-A)A = \frac{3}{4}I $$
$$ -A^2 = \frac{3}{4}I $$
$$ A^2 = -\frac{3}{4}I $$
Step 5: Evaluate the given statements:
1. **$A$ is skew-symmetric matrix of odd order.**
If $A$ is a skew-symmetric matrix of odd order, then $\det(A) = 0$.
From $A^T A = \frac{3}{4}I$, we have $\det(A^T A) = \det\left(\frac{3}{4}I\right)$.
$\det(A^T)\det(A) = \left(\frac{3}{4}\right)^n \det(I)$, where $n$ is the order of $A$.
$(\det(A))^2 = \left(\frac{3}{4}\right)^n$.
If $n$ is odd, $\det(A)=0$, which implies $0 = (\frac{3}{4})^n$, a contradiction.
Therefore, $A$ cannot be of odd order. This statement is incorrect.
2. **$A^2 = \frac{3}{4}I$.**
From Step 4, we found $A^2 = -\frac{3}{4}I$. This statement is incorrect.
3. **$A$ is skew-symmetric matrix of even order.**
From Step 3, $A$ is skew-symmetric. From the analysis in statement 1, $A$ must be of even order. This statement is correct.
4. **$A$ is orthogonal.**
For $A$ to be orthogonal, $A^T A = I$.
From Step 2, we found $A^T A = \frac{3}{4}I$.
Since $\frac{3}{4}I \neq I$, $A$ is not orthogonal. This statement is incorrect.
The incorrect statements are 1, 2, and 4.
Correct Answer: 1, 4