SATHEE: Matrices and Determinants 3 Question 6

Matrices and Determinants 3 Question 6

7. 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 $B B^{T}$ is equal to

(a) $I + B$

(b) $I$

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

Show Answer

Answer:

Correct Answer: 7. (b)

Solution:

PLAN: Use the following properties of transpose

$(A B)^{T} = B^{T} A^{T}, {(A^{T})}^{T} = A$ and $A^{- 1} A = I$ and simplify. If $A$ is non-singular matrix, then $| A | \neq 0$ .

Given, $A A^{T} = A^{T} A$ and $B = A^{- 1} A^{T}$

$\begin{aligned} B B^{T} & = (A^{- 1} A^{T}) {(A^{- 1} A^{T})}^{T} \\ = A^{- 1} A^{T} A {(A^{- 1})}^{T} [∵ (A B)^{T} = B^{T} A^{T}] \\ = A^{- 1} A A^{T} {(A^{- 1})}^{T} [∵ A A^{T} = A^{T} A] \\ = I A^{T} {(A^{- 1})}^{T} [∵ A^{- 1} A = I] \\ = A^{T} {(A^{- 1})}^{T} = {(A^{- 1} A)}^{T} \\ = I^{T} = I \end{aligned}$

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Matrices and Determinants 3 Question 6

7. If A is a 3×3 non-singular matrix such that AAT=ATA and B=A−1AT, then BBT is equal to

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7. 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 $B B^{T}$ is equal to