How To Find Minor Of A Matrix

You're staring at a 4×4 matrix on your problem set. Still, the question asks for the determinant. You know the drill — expand along a row or column, find the minors, apply the signs, sum it up. But then you pause. What exactly is a minor again? Is it the determinant of the submatrix? The submatrix itself? And wait — does the sign come with* the minor or after*?

If that moment sounds familiar, you're not alone. Minors are one of those linear algebra concepts that everyone uses but few can define cleanly on the spot. Let's clear it up once and for all.

What Is a Minor of a Matrix

A minor is a determinant. That's the short version. But the longer version matters because textbooks and professors sometimes use the word loosely.

Given a square matrix A of size n×n, pick any entry aᵢⱼ — that's the element in row i, column j. Delete row i and column j entirely. What remains is an (n-1)×(n-1) submatrix. The minor of aᵢⱼ, usually written Mᵢⱼ, is the determinant of that submatrix.

Not the submatrix itself. The determinant of it.

Minor vs. Cofactor — The Distinction That Trips Everyone

Here's where the confusion lives. The minor Mᵢⱼ is just the determinant. The cofactor Cᵢⱼ (or Aᵢⱼ in some books) is the minor with a sign attached:

Cᵢⱼ = (-1)ⁱ⁺ʲ Mᵢⱼ*

That's it. The sign alternates in a checkerboard pattern starting with + in the top-left. So the minor is always non-negative in spirit — it's just a number. The cofactor carries the sign information needed for determinant expansion and adjugate matrices.

Some texts call Mᵢⱼ the "minor" and Cᵢⱼ the "signed minor.If a formula has (-1)ⁱ⁺ʲ explicitly written out, the thing it multiplies is the minor. Consider this: " Others use "minor" interchangeably for both. When in doubt, check whether the sign is included. If the sign is already baked in, you're looking at a cofactor.

Principal Minors and Leading Principal Minors

Two special flavors show up in advanced topics like Sylvester's criterion for positive definiteness:

A principal minor is the determinant of a submatrix formed by deleting the same* set of rows and columns (e.g., keep rows 1,3 and columns 1,3).
A leading principal minor is the determinant of the top-left k×k submatrix for k = 1, 2, ..., n.

These aren't just vocabulary — they're the gatekeepers for classifying quadratic forms and checking matrix definiteness without computing eigenvalues.

Why It Matters / Why People Care

Minors aren't an end in themselves. They're the connective tissue of matrix theory. Here's where they actually show up:

Determinant expansion (Laplace expansion). The determinant of an n×n matrix is a sum of products of entries and their cofactors — which means minors. Every recursive determinant algorithm bottoms out at 2×2 or 3×3 minors.

Matrix inverse via the adjugate. The inverse A⁻¹ = (1/det A) adj(A). The adjugate is the transpose of the cofactor matrix. Every entry of adj(A) is a cofactor — so every entry is a signed minor. If you're computing an inverse by hand (or debugging a numerical routine), you're swimming in minors.

Cramer's rule. Solving Ax = b* by determinants? Each variable xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i with b. Computing those determinants means more minors.

Rank determination. The rank of a matrix is the size of the largest non-zero minor. This is the theoretical definition. In practice you'd row-reduce, but the minor-based definition explains why rank works the way it does.

Multivariable calculus. The Jacobian determinant — essential for change of variables in integrals — is built from minors of the derivative matrix. The Hessian matrix's leading principal minors classify critical points.

You don't compute minors for fun. You compute them because every structural property of a matrix eventually reduces to them.

Want to learn more? We recommend number of protons neutrons and electrons in beryllium and does hypobromous acid have hydrogen bonding for further reading.

How to Find a Minor — Step by Step

Let's walk through it with a concrete example. No abstract fluff.

Step 1: Identify the Target Entry

Say we have matrix A:

A = [ 2  -1   3 ]
    [ 0   4  -2 ]
    [ 1   5   1 ]

We want the minor M₂₃ — the minor of the entry in row 2, column 3. That entry is -2.

Step 2: Delete the Row and Column

Cross out row 2 and column 3 entirely. What's left?

[ 2  -1 ]
[ 1   5 ]

This 2×2 matrix is the submatrix. It is not the minor. The minor is its determinant.

Step 3: Compute the Determinant of the Submatrix

For a 2×2 matrix [ a b ; c d ], the determinant is ad - bc*.

So M₂₃ = (2)(5) - (-1)(1) = 10 + 1 = 11.

That's it. M₂₃ = 11.

If you needed the cofactor C₂₃, you'd apply the sign: (-1)²⁺³ = (-1)⁵ = -1. So C₂₃ = -11.

Larger Matrices: The Recursive Nature

For a 4×4 matrix, each minor is a 3×3 determinant. For a 5×5, each minor is a 4×4 determinant — which itself expands into 3×3 minors. This recursion is why hand computation explodes factorially.

Let's do a 4×4 example to show the pattern:

B = [ 1  2  3  4 ]
    [ 5  6  7  8 ]
    [ 9  1  2  3 ]
    [ 4  5  6  7 ]

Find M₁₄ (row 1, column 4). Delete row 1, column 4:

[ 5  6  7 ]
[ 9  1  2 ]
[ 4  5  6 ]

Now compute this 3×3 determinant. Pick a row or column — say, the first row:

M₁₄ = 5 × det([1 2

; 3] - 6 × det([9 2; 4 6]) + 7 × det([9 1; 4 5])

Wait, we missed the 4th column's role in the expansion. Let's use the standard Laplace expansion along the first row of our submatrix:

$5 \times \det\begin{bmatrix} 1 & 2 \ 5 & 6 \end{bmatrix} = 5 \times (6 - 10) = -20$
$-6 \times \det\begin{bmatrix} 9 & 2 \ 4 & 6 \end{bmatrix} = -6 \times (54 - 8) = -264$
$7 \times \det\begin{bmatrix} 9 & 1 \ 4 & 5 \end{bmatrix} = 7 \times (45 - 4) = 301$

Summing them up: $-20 - 264 + 301 = 17$. Thus, $M_{14} = 17$.

The Complexity Trap

As demonstrated, calculating a single minor in a large matrix is a recursive nightmare. Instead, we use LU Decomposition or QR Decomposition. This is why, in modern computational linear algebra, we almost never use minors for large-scale problems. Because of that, if you were to calculate every minor for a $10 \times 10$ matrix to find the adjugate, you would be calculating 100 different $9 \times 9$ determinants. These methods achieve the same results (inverses, determinants, and solving systems) in polynomial time, avoiding the factorial "explosion" that makes manual minor calculation impossible for anything larger than a $4 \times 4$.

Conclusion

Minors are the "DNA" of a matrix. On the flip side, while they are computationally inefficient for large-scale data science or engineering simulations, they are conceptually indispensable. Day to day, whether you are determining if a system of equations has a unique solution, finding the rate of change in a multivariable field, or calculating the rank of a transformation, you are ultimately relying on the power of the minor. Consider this: they provide the bridge between the raw numbers in a grid and the deep geometric and algebraic properties of the linear transformation that matrix represents. Understanding them is not just an exercise in arithmetic; it is an exercise in understanding the very structure of linear space.