Matrices & Determinants
Matrix multiplication
Grade Class 12
Question:
Let A = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}. If M and N are two matrices given by M = \sum_{k=1}^{10} A^{2k} and N = \sum_{k=1}^{10} A^{2k-1} then MN^2 is <br> (1) a non-identity symmetric matrix <br> (2) a skew-symmetric matrix <br> (3) neither symmetric nor skew-symmetric <br> (4) an identity matrix
(1) a non-identity symmetric matrix
(2) a skew-symmetric matrix
(3) neither symmetric nor skew-symmetric
(4) an identity matrix
Step-by-Step Solution
Key Concept: Calculate powers of A. A^2 = -4I, A^3 = -4A, A^4 = 16I. Use geometric series sum formula for M and N.
Given A = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}. A^2 = \begin{pmatrix} -4 & 0 \\ 0 & -4 \end{pmatrix} = -4I. Then A^3 = -4A, A^4 = 16I, A^5 = 16A, A^6 = -64I. M = A^2 + A^4 + ... + A^{20} = (-4I) + (16I) + ... + (-4)^{10}I. This is a GP with 10 terms. N = A + A^3 + ... + A^{19} = A(I + A^2 + ... + A^{18}) = A(I - 4I + 16I - ...). Since A^2 = -4I, A^4 = 16I, etc., the sum of powers follows a pattern. MN^2 calculation leads to a symmetric matrix.
Correct Answer: 1