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Skill: Determine the adjugate of a matrix

Explanation and Free Practice Resources

For a small square matrix, the adjugate is the transpose of its cofactor matrix: each cofactor is found from the determinant of the corresponding minor, with signs alternating by position. This construction gives A adj⁡(A)=det⁡(A)IA\,\operatorname{adj}(A)=\det(A)I and supports finding an inverse when the determinant is nonzero; the treatment is limited to low-order matrices, not general proofs or advanced extensions to arbitrary matrix settings.

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Detailed Explanation: Determine the adjugate of a matrix

To find the adjugate of a square matrix:

  1. Find each minor by deleting the row and column of an entry, then taking the determinant of what remains.
  2. Apply the alternating signs
(+−+−+−+−+) \begin{pmatrix} +&-&+\\ -&+&-\\ +&-&+ \end{pmatrix}

to get the cofactor matrix. 3. Transpose the cofactor matrix. The result is the adjugate.

For example, find the adjugate of

A=(120011201).A=\begin{pmatrix} 1&2&0\\ 0&1&1\\ 2&0&1 \end{pmatrix}.

Find the cofactors one at a time. For instance,

C12=−det⁡(0121)=−(−2)=2,C23=−det⁡(1220)=−(−4)=4.C_{12}=-\det\begin{pmatrix}0&1\\2&1\end{pmatrix} =-(-2)=2, \qquad C_{23}=-\det\begin{pmatrix}1&2\\2&0\end{pmatrix} =-(-4)=4.

Repeating this for each position gives the cofactor matrix

(12−2−2142−11).\begin{pmatrix} 1&2&-2\\ -2&1&4\\ 2&-1&1 \end{pmatrix}.

Now transpose it:

adj⁡(A)=(1−2221−1−241).\operatorname{adj}(A)= \begin{pmatrix} 1&-2&2\\ 2&1&-1\\ -2&4&1 \end{pmatrix}.

Remember: the adjugate is the transpose of the cofactor matrix, not the cofactor matrix itself.

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    Matrices - Inverse from Determinant and Adjoint (3x3, without Formula) - Full Matrix

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