How do you find the adjoint of a 2x2 matrix?

For a 2x2 matrix [[a, b], [c, d]], the adjoint is [[d, -b], [-c, a]]. You swap the elements on the main diagonal (a and d), and negate the off-diagonal elements (b and c). This is a direct shortcut that avoids needing to compute cofactors explicitly, though it is equivalent to the general cofactor-then-transpose method.

What is the difference between the adjoint and the inverse of a matrix?

The inverse of a matrix A (written A inverse) equals adj(A) divided by det(A). So the adjoint is the numerator of the inverse formula before dividing by the determinant. If det(A) equals zero, the matrix has no inverse (it is singular), but the adjoint still exists. The adjoint is useful in theoretical contexts and in deriving Cramer's rule. In practice, numerical methods (such as Gaussian elimination) are used to compute inverses of larger matrices.

Adjoint Matrix Calculator

This calculator finds the adjoint (also called the adjugate) of a square matrix, a step used in the standard formula for a matrix inverse and in Cramer's rule. Choose whether you are working with a 2x2 or 3x3 matrix, then enter your values into the matrix grid, which comes pre-filled with example numbers so you can see how it works straight away. As soon as you change a figure, the calculator works out every cofactor by deleting the relevant row and column and applying the checkerboard sign pattern, builds the cofactor matrix, then transposes it to give you adj(A). The results panel shows your original matrix, the cofactor matrix, and the final adjoint side by side, along with the determinant det(A). It then checks its own working by confirming that A multiplied by adj(A) equals det(A) multiplied by the identity matrix, showing a verified pass or a note if the matrix is singular, meaning det(A) equals zero, in which case the adjoint still exists but no inverse does. For 2x2 matrices it also shows the shortcut of swapping the diagonal terms and negating the off-diagonal terms. Use this tool to check homework, to see the cofactor working behind the adjoint method, or as a step before finding the actual inverse with the related Matrix Inverse Calculator. It works with any real-valued 2x2 or 3x3 matrix, including negative and decimal entries.

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Standard method Cofactor expansion and transpose. Works for any real-valued 2x2 or 3x3 matrix.

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What Is the Adjoint of a Matrix?

The adjoint of a square matrix A, written adj(A), is the transpose of its cofactor matrix. It is sometimes called the adjugate to distinguish it from the adjoint defined in inner product spaces (which is the conjugate transpose).

For each element a_ij of A, the cofactor C_ij is:

C_ij = (-1)^i+j M_ij

where M_ij is the minor: the determinant of the submatrix formed by deleting row i and column j from A. The cofactor matrix C has C_ij at position (i, j). The adjoint is then the transpose of C: adj(A)_ij = C_ji.

The 2x2 Shortcut

For a 2x2 matrix A = [[a, b], [c, d]], applying the cofactor-then-transpose rule gives:

Position	Minor	Sign	Cofactor
C₁₁	d	+1	d
C₁₂	c	-1	-c
C₂₁	b	-1	-b
C₂₂	a	+1	a

After transposing, adj(A) = [[d, -b], [-c, a]]. The main diagonal elements swap and the off-diagonal elements are negated.

Worked Example (Default Values)

For A = [[1, 2], [3, 4]]:

Cofactor C₁₁ = (+1)(4) = 4
Cofactor C₁₂ = (-1)(3) = -3
Cofactor C₂₁ = (-1)(2) = -2
Cofactor C₂₂ = (+1)(1) = 1
Cofactor matrix = [[4, -3], [-2, 1]]
adj(A) = transpose of cofactor matrix = [[4, -2], [-3, 1]]

Verification: det(A) = (1)(4) - (2)(3) = 4 - 6 = -2. Then A times adj(A) should equal -2 times I:

[[1,2],[3,4]] x [[4,-2],[-3,1]] = [[(4-6),(-2+2)],[(12-12),(-6+4)]] = [[-2,0],[0,-2]] = -2 x I. Confirmed.

The Adjoint-Inverse Relationship

If det(A) is not zero, the matrix inverse is:

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

This is Cramer's rule in matrix form. If det(A) equals zero, the matrix is singular and has no inverse, but the adjoint is still defined.

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Method: Cofactor expansion (computing each minor by deleting the corresponding row and column, applying the checkerboard sign pattern, then transposing the result). Verification uses the identity A adj(A) = det(A) I.

This calculator works with real numbers. For complex-valued matrices, the adjoint in the Hermitian sense is the conjugate transpose, which is a different operation.