A = [ 1 - 2 0 1 ] , B = [ 5 2 2 1 ] then ( A B A T ) 5 ( A B T A T ) 10 = X then trace of matrix X is ______

Question

$A = [\begin{matrix} 1 & - 2 \\ 0 & 1 \end{matrix}],$ $B = [\begin{matrix} 5 & 2 \\ 2 & 1 \end{matrix}] then {(A B A^{T})}^{5} {(A B^{T} A^{T})}^{10} = X then trace of matrix X is$ ______

Smart Achievers · Accepted Answer

(4)

$A B A^{T} = I \Rightarrow A B B^{T} A^{T} = I$

Solution

Q.
$A = [\begin{matrix} 1 & - 2 \\ 0 & 1 \end{matrix}],$ $B = [\begin{matrix} 5 & 2 \\ 2 & 1 \end{matrix}] then {(A B A^{T})}^{5} {(A B^{T} A^{T})}^{10} = X then trace of matrix X is$ ______

1 4

2 3

3 8

4 2

Ans.
(4)

$A B A^{T} = I \Rightarrow A B B^{T} A^{T} = I$

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Solution

Q. A=[1-201], B=[5221] then (ABAT)5(ABTAT)10=X then trace of matrix X is ______

1 4

2 3

3 8

4 2

Ans. (4) ABAT=I⇒ABBTAT=I

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Q.
$A = [\begin{matrix} 1 & - 2 \\ 0 & 1 \end{matrix}],$ $B = [\begin{matrix} 5 & 2 \\ 2 & 1 \end{matrix}] then {(A B A^{T})}^{5} {(A B^{T} A^{T})}^{10} = X then trace of matrix X is$ ______

Ans.
(4)

$A B A^{T} = I \Rightarrow A B B^{T} A^{T} = I$