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If $A = [\begin{matrix} a & b & c \\ x & y & z \\ p & q & r \end{matrix}], B = [\begin{matrix} q & - b & y \\ - p & a & - x \\ r & - c & z \end{matrix}]$ and if A is invertible. Then which of the following is true?

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|A|=|B|

|A|=−|B|

|A|=−2|B|

|A|=2|B|

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detailed solution

Correct option is B

|B|=q−by−pa−xr−cz _( multiplying _R2 by-1) =−q −b yp −a xr −c zmultiplying C2 by −1 =qbypaxrcz=−paxqbyrcz=−axpbyqczr=−abcxyzpqr=−|A| Hence |A|=−|B|, obviously when |A|≠0,|B|≠0. Also, ∣ adj B|=|B2=(−|A|)2=|A|2

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If A=abcxyzpqr,B=q−by−pa−xr−cz and if A is invertible. Then which of the following is true?

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If $A = [\begin{matrix} a & b & c \\ x & y & z \\ p & q & r \end{matrix}], B = [\begin{matrix} q & - b & y \\ - p & a & - x \\ r & - c & z \end{matrix}]$ and if A is invertible. Then which of the following is true?