First slide

Applications of determinant

Question

$If A is a 3 \times 3 non-singular matrix such that A A^{T} = A^{T} A and B = A^{- 1} A^{T}, then B B^{T} =$

Moderate

A

$I + B$
B

$I$
C

$B^{- 1}$
D

${(B^{- 1})}^{T}$

Solution

$Given, A A^{T} = A^{T} A and B = A^{- 1} A^{T}$
$\begin{array}{l} B B^{T} = (A^{- 1} A^{T}) {(A^{- 1} A^{T})}^{T} \\ = A^{- 1} A^{T} A {(A^{- 1})}^{T} (∵ (A B)^{T} = B^{T} A^{T}) \\ = A^{- 1} A A^{T} {(A^{- 1})}^{T} (∵ A A^{T} = A^{T} A) \end{array}$
$\begin{array}{l} = I A^{T} {(A^{- 1})}^{T} (∵ A^{- 1} A = I) \\ = A^{T} {(A^{- 1})}^{T} = {(A^{- 1} A)}^{T} (∵ (A B)^{T} = B^{T} A^{T}) \\ = I^{T} = I \end{array}$

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