How to construct the classical adjoint matrix of a 3×3 matrix using second-order minor determinants?

Q: How to construct the classical adjoint matrix of a 3×3 matrix using second-order minor determinants?

To construct the classical adjoint matrix ($adj A$), first calculate all nine second-order minor determinants $|A_{ij}|$. Then, apply the alternating sign pattern $(-1)^{i+j}$ to each minor to form the cofactor matrix. Finally, transpose the cofactor matrix. The entry at position $(i,j)$ in the adjugate matrix is the cofactor $C_{ji}$ from the original matrix, meaning the adjugate is the transpose of the signed cofactor matrix.

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How to construct the classical adjoint matrix of $a 3\times 3$ matrix using second-order minor determinants?

To construct the classical adjoint matrix (

adj A

), first calculate all nine second-order minor determinants

|A_{ij}|

. Then, apply the alternating sign pattern

(-1)^{i+j}

to each minor to form the cofactor matrix. Finally, transpose the cofactor matrix. The entry at position

(i,j)

in the adjugate matrix is the cofactor

C_{ji}

from the original matrix, meaning the adjugate is the transpose of the signed cofactor matrix.

Conditions

A is a 3x3 matrix.
All nine second-order minor determinants have been calculated.

Reasoning, step by step

Calculate the minor determinant $|A_{ij}|$ for each position by deleting row i and column j.
Apply the sign $(-1)^{i+j}$ to each minor to get the cofactor $C_{ij}$ .
Arrange the cofactors into a 3x3 matrix.
Transpose this cofactor matrix to obtain the classical adjoint matrix $adj A$ .
Alternatively, directly place the signed minor $(-1)^{i+j}|A_{ji}|$ at position $(i,j)$ in the adjugate matrix.

Example

The screen writes $adj A=\begin{bmatrix}|A_{11}|&-|A_{21}|&|A_{31}|\\-|A_{12}|&|A_{22}|&-|A_{32}|\\|A_{13}|&-|A_{23}|&|A_{33}|\end{bmatrix}$ . Notice that the minor $|A_{21}|$ (row 2, col 1) is placed at position (1,2) with a negative sign.

Common misconceptions

Filling the minors into the adjugate matrix in their original positions without transposing.
Applying the signs but forgetting to transpose the matrix.
Transposing the unsigned minors and then trying to apply signs, which leads to incorrect sign placement.

Watch the explanation

YouTubeFind a Matrix Inverse Using the Classical Adjugate

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Connected concepts

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How to construct the classical adjoint matrix of $a 3\times 3$ matrix using second-order minor determinants?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

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Related questions

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Conditions

Reasoning, step by step

Example

Common misconceptions

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How to construct the classical adjoint matrix of $a 3\times 3$ matrix using second-order minor determinants?