Question-8

Find the adjugate of a general \(3\times3\) diagonal matrix, say \(D\). If \(D=\text{adj}(D)\), what can you say about \(D\)?

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Let \(D=\begin{bmatrix}a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & c \end{bmatrix}\). The cofactor matrix is given by:

\[ \begin{aligned} C & =\begin{bmatrix}bc & 0 & 0\\ 0 & ca & 0\\ 0 & 0 & ab \end{bmatrix} \end{aligned} \]

We see that the cofactor matrix is also diagonal. The adjugate is equal to the transpose of the cofactor matrix. Since \(C\) is diagonal, \(\text{adj}(D)=C\).

If \(D=\text{adj}(D)\), we have:

\[ \begin{aligned} a & =bc\\ b & =ca\\ c & =ab \end{aligned} \]

If any one of \(a,b,c\) is zero, then all three are zero. So we can safely assume that \(a,b,c\neq0\). Plugging the first equation into the second:

\[ \begin{aligned} b & =ca\\ & =bc^{2}\\ b(c^{2}-1) & =0\\ c & =\pm1 \end{aligned} \]

By a similar argument, we get \(a=\pm1\) and \(b=\pm1\). We get the following solutions for \((a,b,c)\):

\[ \begin{aligned} (1,1,1)\\ (1,-1,-1),(-1,1,-1),(-1,-1,1) \end{aligned} \]