The minor of a matrix is the determinant of a smaller submatrix created by deleting one row and one column from the original matrix. Specifically, the (i, j) minor, written Mij, is the determinant you get after removing the i-th row and j-th column. Minors are the building blocks for calculating determinants of larger matrices, finding matrix inverses, and computing cofactors.
How to Calculate a Minor
Start with a square matrix A that is n × n. To find the minor Mij, follow these steps:
- Delete the entire i-th row from the matrix.
- Delete the entire j-th column from the matrix.
- You now have a smaller (n−1) × (n−1) matrix. Take its determinant. That determinant is the minor Mij.
For a 3×3 matrix, each minor is the determinant of a 2×2 submatrix. For a 4×4 matrix, each minor is the determinant of a 3×3 submatrix, and so on. The process always reduces the size by one in each dimension.
A Worked Example
Consider the 3×3 matrix:
A = [ 2, 1, 3 / −1, 2, 1 / −2, 2, 3 ]
To find M11 (the minor for row 1, column 1), delete the first row and first column. The remaining 2×2 matrix is [2, 1 / 2, 3]. Its determinant is (2)(3) − (1)(2) = 4, so M11 = 4.
To find M12 (row 1, column 2), delete the first row and second column. The remaining matrix is [−1, 1 / −2, 3]. Its determinant is (−1)(3) − (1)(−2) = −1, so M12 = −1.
To find M13 (row 1, column 3), delete the first row and third column. The remaining matrix is [−1, 2 / −2, 2]. Its determinant is (−1)(2) − (2)(−2) = 2, so M13 = 2.
Every entry in the matrix has its own minor. A 3×3 matrix has nine minors total, one for each position.
Minors vs. Cofactors
A cofactor is simply a minor with a possible sign change. The cofactor Cij equals (−1)i+j times the minor Mij. That means when i + j is even, the cofactor equals the minor exactly. When i + j is odd, the cofactor is the minor multiplied by −1.
This creates a checkerboard pattern of signs across the matrix:
[ +, −, + / −, +, − / +, −, + ]
In the example above, M11 = 4 and C11 = 4 (since 1+1 = 2, which is even). But M12 = −1 and C12 = −(−1) = 1 (since 1+2 = 3, which is odd). The distinction between minors and cofactors matters whenever you use them in a formula, so it’s worth keeping straight.
Using Minors to Find a Determinant
The main reason minors exist is Laplace expansion, the standard method for computing determinants of matrices larger than 2×2. The algorithm works like this: pick any single row or column, then multiply each element in that row or column by its corresponding cofactor, and add the results together.
Continuing the earlier example, expanding along the first row:
det(A) = (2)(C11) + (1)(C12) + (3)(C13)
det(A) = (2)(4) + (1)(1) + (3)(2) = 8 + 1 + 6 = 15
You can choose any row or column and you’ll get the same answer. Picking a row or column that contains zeros makes the calculation faster, since any term multiplied by zero drops out entirely.
Minors in Matrix Inversion
Minors also play a central role in finding the inverse of a matrix. The process involves three stages. First, you compute the minor for every entry in the matrix, producing a full “matrix of minors.” Second, you apply the checkerboard sign pattern to convert each minor into a cofactor, giving you the “cofactor matrix.” Third, you transpose the cofactor matrix (swap its rows and columns) to get what’s called the adjugate matrix. Dividing each entry of the adjugate by the determinant of the original matrix gives you the inverse.
This method works for any invertible square matrix, though for large matrices it becomes computationally expensive. For a 3×3 matrix you need nine 2×2 determinants. For a 4×4, you need sixteen 3×3 determinants, each of which requires its own set of minors.
Higher-Order Minors
The minors described so far are called “first minors” because they involve deleting exactly one row and one column. But the concept generalizes. You can delete multiple rows and multiple columns at once to create a smaller submatrix, and the determinant of that submatrix is called a higher-order minor. For instance, deleting the second and third rows and the sixth column of a matrix produces a submatrix whose determinant is a minor of higher order.
One practical use of these generalized minors is determining the rank of a matrix. The rank equals the size of the largest square submatrix that has a nonzero determinant. If the largest nonzero minor you can find comes from a 3×3 submatrix, the matrix has rank 3, regardless of how many rows and columns the full matrix contains.
Special Cases
For a 2×2 matrix, the minors are especially simple. Deleting one row and one column leaves a single entry, so each minor is just that one remaining number. If your matrix is [a, b / c, d], then M11 = d, M12 = c, M21 = b, and M22 = a.
The minor of order zero (the case where you haven’t deleted anything meaningful) is conventionally defined to be 1. This might seem odd, but it keeps certain recursive formulas consistent.
Minors are only defined for square matrices or square submatrices extracted from rectangular ones. You need a square array to take a determinant, so the submatrix you create must always have equal numbers of rows and columns.