Quick answer

The cofactor matrix has entries C_ij = (−1)^(i+j) M_ij. The adjoint is adj(A) = (C_ij)^T.

Formula

  • M_ij = minor determinant
  • C_ij = (−1)^(i+j) M_ij
  • adj(A) = (C_ij)^T

Introduction

Students search for a cofactor matrix calculator when homework piles up minors. On this site, the adjoint tool implements those cofactors internally and displays the transposed result adj(A).

Read what is an adjoint matrix for the big picture, then use the Adjoint Matrix Calculator to verify cofactor logic numerically after you build at least one cofactor table by hand.

The adjoint matrix calculator article explains each input field and how to reset the grid when you want to test many matrices in a row.

Cofactor matrix vs adjoint

The cofactor matrix C stacks signed minors in place. The adjoint transposes that stack: adj(A) = C^T. Swapping the two is a frequent grading mistake.

Minors are determinants of smaller submatrices. For a 3×3 matrix, each minor is a 2×2 determinant you can compute with the same rules you learned earlier.

The checkerboard sign pattern starts positive at (1,1) if you use the (−1)^(i+j) rule. Some books draw a literal plus-minus grid; either aid is fine if you stay consistent.

Row or column expansion is often the fastest way to find det(A) once cofactors are available for one line.

Practice minors on 2×2 blocks inside 3×3 problems before you jump to 4×4, where each minor is itself a 3×3 determinant.

Cofactor formulas

  • M_ij = det(A with row i, column j removed)
  • C_ij = (−1)^(i+j) M_ij
  • Row expansion: det(A) = Σ_j a_1j C_1j (along row 1)
  • adj(A) = (C_ij)^T

When you only need one entry of A⁻¹, Cramer rule uses selected cofactors; the full adjoint is the systematic version for all entries.

Assemble C without transposing first, then transpose once to obtain adj(A).

Can cofactors be zero? Yes. If a minor is zero, the cofactor is zero regardless of sign.

Enter A on the home calculator and confirm that transposing your C matches the returned adjoint.

Cofactor workflow

  1. Draw the sign chart. Sketch a small checkerboard for your matrix size before computing any numbers.
  2. Compute minors systematically. Move row by row or column by column so you do not skip a position.
  3. Assemble C without transposing. Write the cofactor matrix exactly as C_ij before you take the transpose for adj(A).
  4. Transpose to adj(A). Only after C is complete, transpose to match the classical adjoint definition.
  5. Verify with the tool. Compare transposed C with calculator output entry by entry.

Cofactor table sample

For A = [[4, 0], [1, −2]], M_11 = −2 and C_11 = −2. M_12 = 1 and C_12 = −1. Continue for the second row.

Cofactor matrix C = [[−2, −1], [0, 4]]. Transpose to get adj(A) = [[−2, 0], [−1, 4]].

Compare with calculator output to ensure your sign chart matched (−1)^(i+j) on every position.

Reuse one row of cofactors from this table to expand det(A) without recomputing every minor from scratch.