What is adj A for matrix A?

Question 47

EasyOriginal question

What is $\operatorname{adj} A$ for $A = \begin{bmatrix}4 & 3 \\ 1 & 2\end{bmatrix}$ ?

\begin{bmatrix}2 & -1 \\ -3 & 4\end{bmatrix}

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\begin{bmatrix}2 & 3 \\ 1 & 4\end{bmatrix}

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\begin{bmatrix}-2 & 3 \\ 1 & -4\end{bmatrix}

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\begin{bmatrix}2 & -3 \\ -1 & 4\end{bmatrix}

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Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

For a 2 × 2 matrix, swap the two diagonal elements and change the signs of the other two.
$\operatorname{adj} A = \begin{bmatrix}2 & -3 \\ -1 & 4\end{bmatrix}$ . Check: $A(\operatorname{adj} A) = \begin{bmatrix}5 & 0 \\ 0 & 5\end{bmatrix} = |A|I$ .

Worth noting

$\begin{bmatrix}2 & -1 \\ -3 & 4\end{bmatrix}$ is the matrix of cofactors itself; the adjoint is its transpose.

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