Cofactors, determinants, and adjugates

Let A be an n × n matrix over a field F. The cofactor of an element Aij is the matrix formed by removing the ith row and jth column, denoted A[i, j].

This terminology is less than ideal. The matrix just described is called the cofactor of Aij, but it would more accurately be called the cofactor of (i, j) because it does not depend on the matrix entry Aij per se but rather its position in the matrix A. For example, if someone asked you what the cofactor of 7 is, you’d have to say “That depends. Where is the 7 located?”. But we will stick to standard terminology.

(I wrote a post about a similar phenomenon in statistical notation, but I can’t find it this morning. You can’t answer questions like “What is p(0.5)?” in statistics by without first asking “If you were to write 0.5 using a variable, would you write it as x, y, or θ?”)

You can compute the determinant of a matrix by taking the alternating sum of elements of a row (or column) multiplied by the determinants of their cofactors.

\det A = \sum_{j=1}^n (-1)^{i+j} A_{i,j} \det A_{[i,j]}

This is essentially Cramer’s rule. It suggests a recursive procedure for computing determinants which is theoretically useful, but only computationally useful for very small matrices.

The adjugate of A, written adj(A), is the transpose of the matrix formed by replacing the (i, j) element of A with an alternating sign times its cofactor [1].

\text{adj}(A)_{ij} = (-1)^{i+j} \det A[j,i]

As you might guess, the Mathematica command to compute the adjugate is Adjugate.

Note that “adjugate” is not a typo for “adjoint.” The adjoint of A (now) means something different. What we now call the adjugate was sometimes in the past called the adjoint. What we now call the adjoint is the (conjugate) transpose.

Here’s an application of the adjugate. Let A be a 3 × 3 matrix. Then the characteristic polynomial of A is

x^3 - (\text{tr}\, A) x^2 + \text{tr}\,\left(\text{adj}(A)\right)x - \det A

where tr denotes the trace, the sum of the diagonal elements of a matrix.

Since the trace only depends on the diagonal, you only need to compute the diagonal of a adjugate to use the equation above for the characteristic polynomial. And the alternating terms defining the adjugate are all positive on the diagonal. So the trace of the adjugate is the sum of the determinants of the cofactors of the diagonal: det A11 + det A22 + det A33.

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