Skew-Symmetric Matrix

To find the least value of $$\det(A)$$, we use the properties of skew-symmetric matrices and substitute them into the updated matrix equation.

Step 1: Understand the property of matrix P

The given condition is:

$$P + P^T = 0 \implies P^T = -P$$

This means $$P$$ is a skew-symmetric matrix of order 2. Any real skew-symmetric matrix of order 2 can be expressed as:

$$P = \begin{pmatrix} 0 & k \\ -k & 0 \end{pmatrix}$$

where $$k$$ is a real number.

Step 2: Express matrix A in terms of P

From the second equation, isolate matrix $$A$$:

$$A + 2P = 3I$$

$$A = 3I - 2P$$

Step 3: Construct matrix A

Substitute the identity matrix $$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ and the matrix $$P$$ into the expression for $$A$$:

$$A = 3\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} - 2\begin{pmatrix} 0 & k \\ -k & 0 \end{pmatrix}$$

$$A = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} - \begin{pmatrix} 0 & 2k \\ -2k & 0 \end{pmatrix}$$

$$A = \begin{pmatrix} 3 & -2k \\ 2k & 3 \end{pmatrix}$$

Step 4: Calculate the determinant of A

Evaluate $$\det(A)$$:

$$\det(A) = (3)(3) - (-2k)(2k)$$

$$\det(A) = 9 + 4k^2$$

Step 5: Find the minimum value

Since $$k$$ is a real number, its square is always non-negative ($$k^2 \ge 0$$), which means:

$$4k^2 \ge 0$$

Therefore:

$$\det(A) \ge 9$$

The least value of $$\det(A)$$ is $$9$$, which occurs when $$k = 0$$

Let $$X = A + B$$ and $$Y = A - B$$.

Taking the transpose of $$X$$:

$$X^T = (A + B)^T = A^T + B^T = A - B = Y$$

Taking the transpose of $$Y$$:

$$Y^T = (A - B)^T = A^T - B^T = A - (-B) = A + B = X$$

Therefore, $$X^T = Y$$ and $$Y^T = X$$.

2. Analyzing Statement S1 ($$P$$)

The matrix $$P$$ is defined as $$P = Y^{2023}X^{2023} - X^{2023}Y^{2023}$$.

To check if it is skew-symmetric, find $$P^T$$:

$$P^T = (Y^{2023}X^{2023})^T - (X^{2023}Y^{2023})^T$$

Using the property $$(MN)^T = N^T M^T$$:

$$P^T = (X^T)^{2023}(Y^T)^{2023} - (Y^T)^{2023}(X^T)^{2023}$$

Substitute $$X^T = Y$$ and $$Y^T = X$$:

$$P^T = Y^{2023}X^{2023} - X^{2023}Y^{2023}$$ $$P^T = P$$

Since $$P^T = P$$, matrix $$P$$ is symmetric, not skew-symmetric.

Result: S1 is False.

3. Analyzing Statement S2 ($$Q$$)

Substitute $$A^4 + B^4 = A$$ into the expression for $$Q$$:

$$Q = (A + B)^T A (A - B) + (A - B)^T A (A + B)$$

Using the transpose relationships $$X^T = Y$$ and $$Y^T = X$$:

$$Q = Y A Y + X A X$$

Now, find $$Q^T$$:

$$Q^T = (Y A Y)^T + (X A X)^T$$ $$Q^T = Y^T A^T Y^T + X^T A^T X^T$$

Substitute $$Y^T = X$$, $$X^T = Y$$, and $$A^T = A$$:

$$Q^T = X A X + Y A Y$$ $$Q^T = Q$$

Since $$Q^T = Q$$, matrix $$Q$$ is symmetric.

Result: S2 is True.

Conclusion

• S1 is False (P is symmetric).
• S2 is True (Q is symmetric).

Correct Option: C (Only S2 is true)

Let the given matrix expression be represented as $$C = A B + B A$$. Taking the transpose on both sides and applying the reversal law for matrix multiplication yields:
$$C^T = (A B + B A)^T = (A B)^T + (B A)^T = B^T A^T + A^T B^T$$

Substituting the given properties $$A^T = A$$ and $$B^T = -B$$ into the expression simplifies it to:
$$C^T = (-B)(A) + (A)(-B) = -B A - A B = -(A B + B A) = -C$$

This confirms that $$C$$ is a skew-symmetric matrix. Taking the determinant of both sides, and noting that the matrix is of an odd order $$(3 \times 3)$$, gives:
$$\det(C^T) = \det(-C) \implies \det(C) = (-1)^3 \det(C) \implies \det(C) = -\det(C)$$

Rearranging this algebraic relation results in $$2 \det(C) = 0$$, which means $$\det(C) = 0$$.