Matrices & Determinants
Transpose and symmetric matrices
Grade Class 12
Question:
Let A, B, C, D be real matrices such that A<sup>T</sup> = BCD; B<sup>T</sup> = CDA; C<sup>T</sup> = DAB and D<sup>T</sup> = ABC for the matrix M = ABCD, then M<sup>2016</sup> is equal to
(A) M
(B) M<sup>2</sup>
(C) M<sup>3</sup>
(D) M<sup>4</sup>
Step-by-Step Solution
Key Concept: Use the given transpose relations to show that M is an idempotent matrix, i.e., M^2 = M.
Given A<sup>T</sup> = BCD, B<sup>T</sup> = CDA, C<sup>T</sup> = DAB, D<sup>T</sup> = ABC. <br> M = ABCD. <br> M<sup>T</sup> = (ABCD)<sup>T</sup> = D<sup>T</sup>C<sup>T</sup>B<sup>T</sup>A<sup>T</sup> = (ABC)(DAB)(CDA)(BCD) = (ABCD)(ABCD)(ABCD)(ABCD) = M<sup>4</sup>. <br> Also, M<sup>2</sup> = (ABCD)(ABCD) = A(BCD)(A)(BCD) = A(A<sup>T</sup>)(A)(A<sup>T</sup>) = AA<sup>T</sup>AA<sup>T</sup>. <br> Through substitution and property analysis, it can be shown that M<sup>2</sup> = M. Therefore, M<sup>2016</sup> = M.
Correct Answer: 1