Adjoint Matrix Calculator

Calculate the adjoint (adjugate), cofactor matrix, determinant, and inverse of a 2x2 or 3x3 matrix with step-by-step solutions.

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Results

Determinant
-2

Cofactor Matrix

Adjoint (Adjugate) Matrix

Inverse Matrix (A-1 = adj(A) / det(A))

Step-by-Step Solution

Understanding the Adjoint (Adjugate) Matrix

The adjoint (also called adjugate) of a square matrix A, denoted adj(A), is the transpose of the cofactor matrix of A. It plays a crucial role in linear algebra, particularly in computing the inverse of a matrix using the formula A-1 = adj(A) / det(A), provided the determinant is non-zero.

Cofactors and Minors

For each element aij of a matrix, the minor Mij is the determinant of the submatrix formed by deleting row i and column j. The cofactor Cij is the minor multiplied by (-1)i+j, which creates the "checkerboard" sign pattern:

For a 3x3 matrix, the sign pattern is:

+ - +
- + -
+ - +

Computing the Adjoint

  1. Calculate each cofactor Cij for every element in the matrix.
  2. Arrange these cofactors in a matrix (the cofactor matrix).
  3. Transpose the cofactor matrix (swap rows and columns) to get the adjoint.

2x2 Adjoint

For [[a,b],[c,d]], the adjoint is [[d,-b],[-c,a]].

adj = [[d,-b],[-c,a]]

Determinant

For 2x2: ad - bc. For 3x3: cofactor expansion along any row or column.

det(A) = ad - bc

Inverse Matrix

The inverse exists only when det(A) is non-zero.

A^-1 = adj(A) / det(A)

Cramer's Rule

Uses determinants and the adjoint to solve systems of linear equations.

x = det(Ax) / det(A)

Relationship to the Inverse

The fundamental relationship between a matrix, its adjoint, and its determinant is: A * adj(A) = det(A) * I, where I is the identity matrix. This means that if det(A) is not zero, we can solve for the inverse: A-1 = adj(A) / det(A). Each element of the inverse is the corresponding element of the adjoint divided by the determinant.

Cramer's Rule Connection

Cramer's rule provides a method for solving systems of linear equations using determinants. For a system Ax = b, each variable xi can be found by replacing the i-th column of A with b and computing the determinant of the modified matrix, divided by det(A). This method is computationally expensive for large matrices but provides a direct formula and is useful for theoretical analysis and small systems.

Properties of the Adjoint

  • adj(AB) = adj(B) * adj(A) (note the reversed order).
  • adj(AT) = (adj(A))T.
  • adj(kA) = kn-1 * adj(A) for an n x n matrix.
  • det(adj(A)) = (det(A))n-1 for an n x n matrix.
  • For a 2x2 matrix, adj(adj(A)) = A.