If A is a skew-symmetric matrix of order 2 and B, Carematrices 1 42 9 and 9 −4−2 1 respectively. ThenA3BC+A5B2C2+A7B3C3+…+A2n+1BnCn, is

A
a symmetric matrix
B
a skew-symmetric matrix
C
an identity matrix
D
none of these

Solution:

We observed that $B C = I = C B$

$B^{n} C^{n} = C^{n} B^{n} = I$ for all $n \in N$

Let $X = A^{3} B C + A^{5} (B^{2} C^{2}) + A^{7} (B^{3} C^{3}) + \dots + A^{2 n + 1} B^{n} C^{n}$ . Then

$X = A^{3} I + A^{5} I + A^{7} I + \dots + A^{2 n + 1} I$

$\begin{array}{l} \Rightarrow X = A^{3} + A^{5} + A^{7} + \dots + A^{2 n + 1} \\ \Rightarrow X^{T} = {(A^{3} + A^{5} + A^{7} + \dots + A^{2 n + 1})}^{T} \\ \Rightarrow X^{T} = {(A^{3})}^{T} + {(A^{5})}^{T} + {(A^{7})}^{T} + \dots + {(A^{2 n + 1})}^{T} \\ \Rightarrow X^{T} = {(A^{T})}^{3} + {(A^{T})}^{5} + {(A^{T})}^{7} + \dots + {(A^{T})}^{2 n + 1} \\ \Rightarrow X^{T} = (- A)^{3} + (- A)^{5} + (- A)^{7} + \dots + (- A)^{2 n + 1} [∵ A^{T} = - A] \\ \Rightarrow X^{T} = - X \Rightarrow X is a skew-symmetric matrix. \end{array}$

If A is a skew-symmetric matrix of order 2 and B, Care
matrices $[\begin{array}{ll} 1 4 \\ 2 9 \end{array}]$ and $[\begin{array}{rr} 9 - 4 \\ - 2 1 \end{array}]$ respectively. Then
$A^{3} B C + A^{5} (B^{2} C^{2}) + A^{7} (B^{3} C^{3}) + \dots + A^{2 n + 1} B^{n} C^{n}$ , is

Solution:

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