Question

Which of the following statements is/are TRUE regarding matrices?

• A. Every square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.
• B. The determinant of a square matrix is same as that of its transpose.
• C. The product of a square matrix and its inverse results in the identity matrix.
• D. Eigenvalues of a square matrix are always real and positive.

Hint:

Test the eigenvalue claim with a simple 2x2 matrix before trusting it as a general rule.

Answer 1

The Correct Option is A, B, C

Step 1: Check statement (A).
Any square matrix \(A\) can be written as \(A = \dfrac{1}{2}(A + A^{T}) + \dfrac{1}{2}(A - A^{T})\). The first part is symmetric, since \((\tfrac{1}{2}(A+A^T))^T = \tfrac{1}{2}(A+A^T)\), and the second part is skew-symmetric, since \((\tfrac{1}{2}(A-A^T))^T = -\tfrac{1}{2}(A-A^T)\). This split always works, so (A) is TRUE.

Step 2: Check statement (B).
A standard determinant property states \(\det(A) = \det(A^{T})\) for any square matrix, because expanding along rows or columns gives the same value either way. So (B) is TRUE.

Step 3: Check statement (C).
By definition, the inverse \(A^{-1}\) of an invertible square matrix \(A\) is exactly the matrix satisfying \(A A^{-1} = A^{-1}A = I\). So (C) is TRUE by definition, whenever the inverse exists.

Step 4: Check statement (D).
This is not true in general. The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) has eigenvalues \(\pm i\), which are complex, and \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) has eigenvalues \(1\) and \(-1\), one of which is negative. Real, positive eigenvalues are only guaranteed for special cases like symmetric positive definite matrices. So (D) is FALSE.

Final Answer:
Statements A, B and C hold for every square matrix, while D fails in general. \[ \boxed{\text{A, B, C}} \]