If A = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 and B = - 4 - 3 - 3 1 0 1 4 4 3 are two matrices, then the value of the determinant A + A 2 B 2 + A 3 + A 4 B 4 + . . . . . . . . . 20 t e r ms is

Question

If A = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 and B = - 4 - 3 - 3 1 0 1 4 4 3 are two matrices, then the value of the determinant A + A 2 B 2 + A 3 + A 4 B 4 + . . . . . . . . . 20 t e r ms is, 20 3, 2 20 3, - 20 3, 0

MARKS App · Accepted Answer

A 2 = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 2 - 2 - 4 - 1 3 4 1 - 2 - 3 = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 = A B 2 = - 4 - 3 - 3 1 0 1 4 4 3 - 4 - 3 - 3 1 0 1 4 4 3 = 1 0 0 0 1 0 0 0 1 = Ι B 2 = B 4 = B 6 = B 8 = . . . . . . . = B 20 = Ι A 2 = A 3 = A 4 = . . . . . . . A 20 = A So, A + A 2 B 2 + A 3 . . . . . . . + A 20 B 20 = A + A + A . . . . . . . + A = 20 A = 20 3 A ….(1) A = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 = 2 - 9 + 8 + 2 3 - 4 - 4 2 - 3 = - 2 - 2 + 4 = 0 Hence, from equation (1), A + A 2 B 2 + A 3 . . . . . . . + A 20 B 20 = 0

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