What is adj A for matrix A?

[1/2, 0, 0; 0, 1/3, 0; 0, 0, 1/4]

Question 50

MediumOriginal question

What is $\operatorname{adj} A$ for $A = \begin{bmatrix}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4\end{bmatrix}$ ?

\begin{bmatrix}\frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{4}\end{bmatrix}

Correct answerYour answer

\begin{bmatrix}6 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 12\end{bmatrix}

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\begin{bmatrix}12 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 6\end{bmatrix}

Correct answerYour answer

\begin{bmatrix}24 & 0 & 0 \\ 0 & 24 & 0 \\ 0 & 0 & 24\end{bmatrix}

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Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

For a diagonal matrix the off-diagonal cofactors are all 0, and the cofactor of each diagonal element is the product of the other two.
So the diagonal of $\operatorname{adj} A$ is $3 \times 4 = 12$ , $2 \times 4 = 8$ and $2 \times 3 = 6$ .

Worth noting

The matrix with $\frac{1}{2}$ , $\frac{1}{3}$ , $\frac{1}{4}$ is $A^{-1}$ , which is $\operatorname{adj} A$ divided by $|A| = 24$ .

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