Adjugate Formula

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Overview: Matrix inversion

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The adjugate formula defines the inverse of an n-by-n square matrix $\mathbf{A}$ as

$\mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\text{adj}(\mathbf{A})$

where $\text{adj}(\mathbf{A})$ is the so called adjugate matrix of $\mathbf{A}$. The adjugate matrix is the transposed of the cofactor matrix:

$\text{adj}(\mathbf{A})=\mathbf{C}(\mathbf{A})^T$

And the cofactor matrix $\mathbf{C}(\mathbf{A})$ is just a matrix where each cell corresponds to the related cofactor:

$\mathbf{C}(\mathbf{A})=\left[\begin{array}{cccc} C_{1,1}(\mathbf{A}) & C_{1,2}(\mathbf{A}) & \cdots & C_{1,n}(\mathbf{A})\\ C_{2,1}(\mathbf{A}) & C_{2,2}(\mathbf{A}) & & C_{2,n}(\mathbf{A})\\ \vdots & & \ddots & \vdots\\ C_{n,1}(\mathbf{A}) & C_{n,2}(\mathbf{A}) & \cdots & C_{n,n}(\mathbf{A}) \end{array}\right]$

So to determine the inverse of an n-by-n square matrix you have to compute the n square cofactors, then transpose the resulting cofactor matrix and divide all the values by the determinant.

Example: inverse of a 4-by-4 matrix using the adjugate formula $\mathbf{A}_e = \left[\begin{array}{cccc} 1 & 2 & 0 & 0\\ 3 & 0 & 1 & 1\\ 0 & 1 & 0 & 0\\ 0 & 0 & 2 & 1 \end{array}\right]$ The calculation of two of the cofactors of $\mathbf{A}_e$ has already been described in the example for minors and cofactors. The resulting cofactor matrix of matrix $\mathbf{A}_e$ is $\mathbf{C}(\mathbf{A}_e)= \left[\begin{array}{cccc} 1 & 0 & 3 & -6\\ 0 & 0 & -1 & 2\\ -2 & 1 & -6 & 12\\ 0 & 0 & 1 & -1 \end{array}\right]$ The transposed of the cofactor matrix which corresponds to the adjugate matrix then appears as $\mathbf{C}(\mathbf{A}_e)^T= \left[\begin{array}{cccc} 1 & 0 & -2 & 0\\ 0 & 0 & 1 & 0\\ 3 & -1 & -6 & 1\\ -6 & 2 & 12 & -1 \end{array}\right]=\text{adj}(\mathbf{A}_e)$ The determinant is computed in the example for a determinant of a matrix and equals 1. So the inverse of $\mathbf{A}_e$ is determined as $\mathbf{A}_e^{-1}=\frac{1}{\det(\mathbf{A}_e)}\text{adj}(\mathbf{A}_e) =\frac{1}{1} \left[\begin{array}{cccc} 1 & 0 & -2 & 0\\ 0 & 0 & 1 & 0\\ 3 & -1 & -6 & 1\\ -6 & 2 & 12 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 0\\ 0 & 0 & 1 & 0\\ 3 & -1 & -6 & 1\\ -6 & 2 & 12 & -1 \end{array}\right]$