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

Each option makes a general claim about square matrices, so the question asks which standard matrix facts hold for every square matrix, and which needs an extra condition not stated here.

1. Every square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix: true. Writing $A = \frac{1}{2}(A+A^T) + \frac{1}{2}(A-A^T)$ always splits any square $A$ into a symmetric part and a skew-symmetric part, with no extra condition needed.
2. The determinant of a square matrix is same as that of its transpose: true. Transposing a matrix does not change the value from the cofactor expansion of its determinant, so $\det(A) = \det(A^T)$ always.
3. The product of a square matrix and its inverse results in the identity matrix: true. This is exactly how the inverse is defined; whenever $A^{-1}$ exists, $AA^{-1} = I$ holds by construction.
4. Eigenvalues of a square matrix are always real and positive: false. This only holds for special matrices, such as symmetric positive definite ones. A general square matrix, for example a simple rotation matrix, can have complex or negative eigenvalues.

So the correct statements are the first three, giving option A, B, C.