The calculator computes the adjugate matrix of a given NxN matrix and uses the result to compute also the inverse matrix. The calculator shows the calculation of every element of the adjugate matrix. The input field N defines the number of rows and columns. The input field digits is for setting the number of displayed digits. With setting of N the related matrix field will be displayed for input of the matrix elements. With selection button 'Compute' the computation of the adjugate and inverse matrix starts. With selection button 'Steps' the elements of the cofactor matrix are shown also.

$A=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1N}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2N}\\ & \vdots \\ {a}_{N1}& {a}_{N2}& \dots & {a}_{NN}\end{array}\right)$

Enter the matrix elements: a_{11}, a_{12}, ...

The adjugate is often used to calculate the inverse of a square matrix. To define the adjugate it is useful to define some terms first.

The minors of a matrix A are formed by crossing out the i-th row and j-th column for each matrix element a_{ij}. The values of these determinants are the minors m_{ij} of matrix A.

The cofactor matrix Cof(A) of a matrix A is formed from the minors by multiplying each minor m_{ij} with a sign (-1)^{i+j}. The elements of the cofactor matrix are thus a^{*}_{ij}=(-1)^{i+j} * m_{ij}.

${a}_{ij}^{*}={\left(-1\right)}^{\left(i+j\right)}{m}_{ij}$

$\mathrm{Cof}\left(A\right)=\left(\begin{array}{cccc}{a}_{11}^{*}& {a}_{12}^{*}& \dots & {a}_{1n}^{*}\\ {a}_{21}^{*}& {a}_{22}^{*}& \dots & {a}_{2n}^{*}\\ & \vdots \\ {a}_{n1}^{*}& {a}_{n2}^{*}& \dots & {a}_{nn}^{*}\end{array}\right)$

The adjugate of matrix of A is defined as follows.

$\mathrm{adj(}A)={\mathrm{Cof}\left(A\right)}^{T}$

Note:

The terms and definitions of the adjugate can easily be misunderstood. In the literature there are different definitions of the adjectives. Sometimes the cofactor matrix is used as adjugate. Furthermore, it should be noted that the adjugate is not the adjugate matrix. The adjugate matrix is for real matrices the same as the transposed matrix and for complex matrices the transposed with conjugated complex elements.

Here is a list of of further useful calculators and sites:

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