Adjoint Matrix
Compute the adjoint (classical adjugate) of a 2x2 matrix step by step. Free online adjoint matrix calculator with formula for matrix algebra.
About This Calculator
This calculator computes the adjoint (classical adjugate) of a 2x2 matrix. For matrix [[a, b], [c, d]], the adjoint is [[d, -b], [-c, a]]. The adjoint is used primarily to compute the inverse of a matrix via A⁻¹ = adj(A) / det(A).
The adjoint is also known as the adjugate or classical adjoint. It is the transpose of the cofactor matrix and an essential concept in linear algebra for matrix inversion and solving systems of linear equations.
Frequently Asked Questions
What is the adjoint of a matrix?
The adjoint (classical adjugate) of a 2x2 matrix [[a, b], [c, d]] is [[d, -b], [-c, a]]. It is the transpose of the cofactor matrix and is used to compute the inverse of a matrix.
How do you find the adjoint of a 2x2 matrix?
For matrix [[a, b], [c, d]], swap a and d, and negate b and c: adj(A) = [[d, -b], [-c, a]]. No determinant calculation is needed for the adjoint itself.
How is the adjoint used to find the inverse?
The inverse of a matrix is A⁻¹ = adj(A) / det(A). First compute the adjoint, then divide each element by the determinant. If det(A) = 0, the matrix is singular and has no inverse.
Is the adjoint the same as the transpose?
No. The adjoint (classical adjugate) is the transpose of the cofactor matrix. The transpose simply swaps rows and columns. They are different concepts in linear algebra.
What is the adjoint of a 3x3 matrix?
For a 3x3 matrix, compute the cofactor for each element, arrange them in a cofactor matrix, then transpose it. Each cofactor is (-1)^(i+j) times the determinant of the 2x2 submatrix.
Is this tool free?
Yes, all calculators on Calculy are completely free to use with no registration required.