Adjoint Matrix Calculator
Calculate adjoint (adjugate) matrix with step-by-step solutions
Matrix Input
Input Matrix A:
Results
Adjoint Matrix adj(A):
Cofactor Matrix C:
Matrix Inverse Formula:
A⁻¹ = (1/det(A)) × adj(A)
A⁻¹ = (1/-2.0000) × adj(A)
Step-by-step solution:
Given 2×2 matrix A:
[ 1.00 2.00] [ 3.00 4.00]
Step 1: Calculate the cofactor matrix C
For each element A[i][j], calculate C[i][j] = (-1)^(i+j) × Minor[i][j]
For 2×2 matrix:
C[0][0] = (-1)^(0+0) × 4 = 4
C[0][1] = (-1)^(0+1) × 3 = -3
C[1][0] = (-1)^(1+0) × 2 = -2
C[1][1] = (-1)^(1+1) × 1 = 1
Cofactor matrix C:
[ 4.00 -3.00] [ -2.00 1.00]
Step 2: Transpose the cofactor matrix to get the adjoint matrix
adj(A) = C^T
Adjoint matrix:
[ 4.00 -2.00] [ -3.00 1.00]
Additional information:
Determinant of A = -2.0000
Since det(A) ≠ 0, the matrix is invertible.
The inverse matrix A^(-1) = (1/det(A)) × adj(A)
Example Calculations
2×2 Matrix Example
Matrix A: [[1, 2], [3, 4]]
Cofactor Matrix: [[4, -3], [-2, 1]]
Adjoint Matrix: [[4, -2], [-3, 1]]
Determinant: -2
3×3 Matrix Example
Matrix A: [[1, 0, 2], [0, 1, 0], [1, 0, 1]]
Steps: Calculate 9 cofactors, arrange in matrix, then transpose
Result: Each element follows the pattern (-1)^(i+j) × Minor[i][j]
Matrix Properties
Adjoint Definition
Transpose of cofactor matrix
adj(A) = C^T
Cofactor Formula
C[i][j] = (-1)^(i+j) × M[i][j]
M is the minor matrix
Inverse Relation
A⁻¹ = adj(A)/det(A)
If det(A) ≠ 0
Calculation Tips
Only square matrices have adjoints
Adjoint = transpose of cofactor matrix
Sign pattern: checkerboard of +/-
Used for matrix inversion
A × adj(A) = det(A) × I
Understanding Adjoint (Adjugate) Matrix
What is an Adjoint Matrix?
The adjoint (or adjugate) matrix of a square matrix A is the transpose of its cofactor matrix. It plays a crucial role in calculating matrix inverses and solving linear systems.
Calculation Steps
- 1.Find the minor of each element by removing its row and column
- 2.Calculate the cofactor using the sign pattern (-1)^(i+j)
- 3.Form the cofactor matrix with all cofactors
- 4.Transpose the cofactor matrix to get the adjoint
Applications
- •Computing matrix inverses
- •Solving linear systems (Cramer's rule)
- •Theoretical matrix analysis
2×2 Matrix Formula
For matrix:
⎣ c d ⎦
Adjoint is:
⎣ -c a ⎦
Important Properties
Matrix Product
A × adj(A) = det(A) × I
Inverse Formula
A⁻¹ = adj(A) / det(A)
Determinant Relation
det(adj(A)) = det(A)^(n-1)
Note
The terms "adjoint" and "adjugate" are often used interchangeably for this concept. In some contexts, "adjoint" may refer to the conjugate transpose, so "adjugate" is sometimes preferred for clarity.