Match the LIST-I with LIST-II
| LIST-I | LIST-II | ||
| A. | All positive integral powers of a symmetric matrix are | I. | Unit matrix. |
| B. | Positive odd integral powers of a skew-symmetric matrix are | II. | Null matrix. |
| C. | All the matrices which are both symmetric and skew-symmetric, are | III. | Skew - Symmetric matrix. |
| D. | All the square matrices whose each diagonal element is one and each non-diagonal element is equal to zero, are | IV. | Symmetric matrix. |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A-IV, B-III, C-II, D-I
(A) If \( A^{T}=A \), then
\( \displaystyle (A^{n})^{T}=(A^{T})^{n} \)
\( \displaystyle =A^{n} \)
So the powers are symmetric, which matches (IV).
(B) If \( A^{T}=-A \) and n is odd, then
\( \displaystyle (A^{n})^{T}=(-A)^{n} \)
\( \displaystyle =-A^{n} \)
So the odd powers are skew-symmetric, which matches (III).
(C) \( A^{T}=A \) and \( A^{T}=-A \) give \( A=-A \), so A = O. These are null matrices, which matches (II).
(D) A square matrix with each diagonal element 1 and every other element 0 is the unit (identity) matrix, which matches (I).
So the matching is (A)-(IV), (B)-(III), (C)-(II), (D)-(I).