If A is a square matrix, then what is {adj} A^{T} - ({adj} A)^{T} equal to?

Correct Answer is : Null matrix{Null matrix}Null matrix

Correct Answer is : Null matrix{Null matrix}Null matrix

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Haryana Police Constable Model Paper 2

Question : 58 of 100

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\operatorname{adj} A^{T} - (\operatorname{adj} A)^{T}

equal to?

Solution:

Concept:

Let A be any square matrix.

(\operatorname{adj} A)^{T} = |A|^{T} (A^{T})^{-1}

(\operatorname{adj} A)^{T}

(|A| A^{-1})^{T}

\operatorname{adj}(A^{T}) = (\operatorname{adj} A)^{T}

Calculations:

Given, A is a square matrix.

Consider,

\operatorname{adj}(A^{T}) - (\operatorname{adj} A)^{T}

= |A^{T}| (A^{T})^{-1} - (|A| A^{-1})^{T}

= |A|^{T} (A^{T})^{-1} - (|A|^{T} (A^{-1})^{T})

= |A|^{T} (A^{-1})^{T} - (|A|^{T} (A^{-1})^{T})

= 0

= Null Matrix

Hence, If A is a square matrix, then

\operatorname{adj} A^{T} - (\operatorname{adj} A)^{T}

equal to null matrix.

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