Question #236795

Given that A1=1det(A)[dbca]A^{-1}=\frac{1}{\textup{det}(A)}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}  with det(A)0\textup{det}(A)\neq 0, consider the following statements:

(i) 1det(A)[dbca]=1det(A)adj(A)\frac{1}{\textup{det}(A)}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}=\frac{1}{\textup{det}(A)}\textup{adj}(A) ;


(ii) A=[acbd]TA=\begin{bmatrix}a&c\\b&d\end{bmatrix}^T ;


(iii) adj(A)=[dcba]T\textup{adj}(A)=\begin{bmatrix}d&-c\\-b&a\end{bmatrix}^T


Which of the above are true?



1
Expert's answer
2021-09-14T06:09:41-0400

We know that for an invertible matrix A

A1=1det(A).adJ(A)....(I)A^{-1}=\frac{1}{det(A)}.adJ(A)....(I)

For a 2×2 matrix A=[abcd]A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}

A1=1det(A)[dbca]......(2)A^{-1}=\frac{1}{det(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}......(2)

We gave

A1=1det(A)[dbca] withA#0A^{-1}=\frac{1}{det(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\space with A\#0


Therefore from equition 1 we can conclude that

A1=1det(A).adJ(A)=1det(A)[dbca].....(2)A^{-1}=\frac{1}{det(A)}.adJ(A)=\frac{1}{det(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.....(2)

Since we have

A1=1det(A)[dbca]A^{-1}=\frac{1}{det(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}


Therefore

A=[abcd]A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}

A=[abcd]T....(4)A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{T}....(4)

Also

adj(A)=[dbca]=[dcba]T....(5)=\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}=\begin{bmatrix} d & -c \\ -b & a \end{bmatrix}^{T}....(5)


Hence, proved that (I)(ii) and (iii) are all true.


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