Let A = [ 0 2 - 3 - 2 0 1 3 - 1 0 ] and B be a matrix such that B B ( I - A ) = I + A Then the sum of the diagonal elements of B T B is equal to______. [2026]

Question

Let $A = [[\begin{matrix} 0 & 2 & - 3 \\ - 2 & 0 & 1 \\ 3 & - 1 & 0 \end{matrix}]]$ and B be a matrix such that $B (I - A) = I + A$ Then the sum of the diagonal elements of $B^{T} B$ is equal to______. [2026]

Smart Achievers · Accepted Answer

(3)

$A^{T} = - A$

$B = (I + A) {(I - A)}^{- 1}$

$B^{T} = ({{(I - A)}^{- 1})}^{T} {(I + A)}^{T}$

$B^{T} = {(I - A^{T})}^{- 1} (I + A^{T})$

$B^{T} = {(I + A)}^{- 1} (I - A)$

$B^{T} B = {(I + A)}^{- 1} (I - A) (I + A) {(I - A)}^{- 1}$

$= {(I + A)}^{- 1} (I + A) (I - A) {(I - A)}^{- 1}$

$= I$

$tr (B^{T} B) = 3$

Solution

Q.
Let $A = [[\begin{matrix} 0 & 2 & - 3 \\ - 2 & 0 & 1 \\ 3 & - 1 & 0 \end{matrix}]]$ and B be a matrix such that $B (I - A) = I + A$ Then the sum of the diagonal elements of $B^{T} B$ is equal to______. [2026]

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Solution

Q. Let A=[02-3-2013-10] and B be a matrix such that B(I-A)=I+A Then the sum of the diagonal elements of BTB is equal to______. [2026]

Ans. (3) AT=-A B=(I+A)(I-A)-1 BT=((I-A)-1)T(I+A)T BT=(I-AT)-1(I+AT) BT=(I+A)-1(I-A) BTB=(I+A)-1(I-A)(I+A)(I-A)-1 =(I+A)-1(I+A)(I-A)(I-A)-1 = I tr(BTB)=3

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Q.
Let $A = [[\begin{matrix} 0 & 2 & - 3 \\ - 2 & 0 & 1 \\ 3 & - 1 & 0 \end{matrix}]]$ and B be a matrix such that $B (I - A) = I + A$ Then the sum of the diagonal elements of $B^{T} B$ is equal to______. [2026]