Cofactor Determinant Calculator

What a cofactor is

A cofactor is a smaller determinant wearing a sign. Pick an entry a(i,j) in a square matrix, cross out row i and column j, and take the determinant of what is left. That leftover value is the minor M(i,j). Multiply it by plus one or minus one and you have the cofactor C(i,j).

In symbols: C(i,j) = (-1)^(i+j) M(i,j). The exponent decides only a sign, so the hard part of any cofactor is still a plain determinant, one size smaller.

The signs form a checkerboard that always starts with plus in the top left corner. The grid below is the pattern for a 4x4 matrix. Each row is the row above it with every sign flipped, so the whole board follows from the top left corner.

One small matrix shows the rule at work. For the 2x2 with rows 3, -2 and -5, 4, the minor M(1,1) is 4 and 1 plus 1 is even, so C(1,1) = 4. The minor M(1,2) is -5 and 1 plus 2 is odd, so C(1,2) = -(-5) = 5. No arithmetic changed between those two steps. Only the sign did.

Expanding a determinant along a row or a column

Laplace expansion in one line

The expansion turns a determinant into a sum along a single line. Fix a row i and add the entry times its cofactor for every column: det(A) = a(i,1) C(i,1) + a(i,2) C(i,2) + ... + a(i,n) C(i,n). Fix a column instead and the same sum runs down the rows. Both produce the same number, and that is not a coincidence. It is the content of Laplace's formula.

Two facts make the method practical. Every one of the 2n lines gives the identical total, so the choice of line is free. And because the cofactors already carry the alternating signs, the terms are added exactly as they come out. There is no separate sign to patch in afterwards, which is where most hand calculations go wrong.

The MIT OpenCourseWare 18.06 course builds the same result from the volume interpretation of a determinant, and the Wolfram MathWorld entry on the determinant states the expansion in the same form. If you want the linear algebra behind it rather than the recipe, the MIT 18.06 materials are free and start from first principles.

Worked example: a 2x2 and a 3x3

Small sizes first. Take the 2x2 matrix with rows 3, -2 and -5, 4. Expanding along the first row gives 3 x 4 = 12 for the first term and (-2) x (-5) = 10 for the second, with a minus between them: 12 - 10 = 2. The familiar ad minus bc rule is a cofactor expansion that nobody bothers to name.

Now the 3x3 matrix with rows 2, -1, 3 and 0, 4, 1 and 5, 2, -1. A 3x3 needs nine minors, one for each entry, and each is a 2x2 determinant. The table lists them by position.

Expanding along row 1 uses the first row of that table. The first term is 2 x (-6) = -12, the second is (-1) x 5 = -5, the third is 3 x (-20) = -60. Adding the three: -12 - 5 - 60 = -77.

The calculator at the top of the page chooses the line with the most zeros, and row 2 of this matrix starts with a zero, so its step list begins with a term that drops out and carries two live terms. The total it prints is the same -77, which is the point of the rule that any line works.

Expanding along column 3 is the check. Its terms are 3 x (-20) = -60. Then 1 x (-9) = -9. Then (-1) x 8 = -8. Written in a different order, they still total -77. When two lines disagree, an arithmetic slip has happened somewhere, and the calculator prints all of them side by side so you can see which one broke.

Worked example: a 4x4 where a whole column is zero

A 4x4 by hand is where zeros pay for themselves. Take the matrix with rows 1, 0, 2, -1 and 3, 0, 0, 5 and 2, 1, 4, -3 and 1, 0, 5, 0. Column 2 reads 0, 0, 1, 0, so three of its four terms vanish before any arithmetic starts.

Only a(3,2) = 1 survives. Its sign is (-1) to the power 5, which is -1, so the whole determinant equals -1 times the minor M(3,2). Deleting row 3 and column 2 leaves the 3x3 with rows 1, 2, -1 and 3, 0, 5 and 1, 5, 0. That minor expands to -25 + 10 - 15, which is -30: the first term is 1 x (0 x 0 - 5 x 5), the second is -2 x (3 x 0 - 5 x 1), and the third is -1 x (3 x 5 - 0 x 1).

Put the sign back and the answer is -1 x (-30) = 30. The calculator works with the signed cofactor directly, so it shows that same term as 1 x 30 = 30 instead of naming the minor first.

Ten entry products did the whole job: nine inside the 3x3 minor and one for the outer multiply. Expanding along row 1 needs three 3x3 minors, so 30 products, and a 4x4 with no zeros anywhere in the chosen line needs the full 40. That gap is the reason the calculator picks the line with the most zeros for you.

Picking the cheapest line

Cost grows fast

Cofactor expansion is not the fastest way to compute a determinant. It is the easiest way to explain one, and for sizes up to 4x4 it is often the quickest by hand. The cost table below counts the entry products a straight expansion needs when every minor is worked out on its own.

The jump from 40 to 205 is the reason nobody expands a 10x10 by hand. Row reduction reaches the same determinant in a time that grows with the cube of the size instead of the factorial, which is what computer algebra systems use. For a 4x4 on paper, expansion with a good choice of line beats a page of row operations.

Choosing well is a one minute exercise. Count the zeros in each row and each column, and expand along the line with the most of them. A line with three zeros out of four reduces a 4x4 to a single 3x3. A line with no zeros at all forces four minors. When two lines tie, prefer the one whose entries are small, since the products stay short.

What the number tells you

A determinant is a single number that answers several questions at once.

A determinant of zero means the matrix is singular. Its rows are linearly dependent, it has no inverse, and a linear system built from it has either no solution or infinitely many. A nonzero determinant means the matrix is invertible and the system has exactly one solution, which is the condition Cramer's rule needs before it can be used.

The absolute value measures volume. The 3x3 matrix whose determinant is -77 maps the unit cube to a shape with volume 77, and the 4x4 above, with determinant 30, scales volumes by 30. The sign records orientation: a negative determinant means the map flips the shape over, the way a mirror does.

Three identities let you shortcut a lot of work. The determinant of a product is the product of the determinants, so det(AB) = det(A) x det(B). The determinant of a transpose equals the determinant of the original, which is why expanding along a column is legal at all. And the determinant of a triangular matrix is just the product of its diagonal, so a matrix with 2, 3 and 4 on the diagonal and anything above it has determinant 2 x 3 x 4 = 24.

Minors, cofactors and the adjugate

The four objects people mix up

Four related objects appear in the same chapter of every linear algebra book, and their names get swapped in a hurry. The table gives the definition and the size of each.

Two mistakes account for most lost marks here. The first is forgetting the transpose: the adjugate is the cofactor matrix flipped over its diagonal, not the cofactor matrix itself. The second is dividing by a zero determinant, which is undefined rather than infinite, because a singular matrix has no inverse at all.

For the 2x2 above with determinant 2, the cofactor matrix is 4, 5 and 2, 3 read by rows, and its transpose swaps the two off diagonal entries. Dividing each entry by 2 gives the inverse, whose top left entry is 2 and whose lower left entry is 2.5. Multiplying the original by that inverse returns the identity, which is the check worth doing once by hand.

Checking an answer without redoing the work

You do not need to recompute every minor to know whether a determinant is right. Four cheap tests catch nearly every mistake.

Expand along a different line. The widget prints the total for all of them, and a mismatch points straight at the term that broke. Scale a row by a number and the determinant should scale by the same number: doubling one row doubles the answer. Swap two rows and the sign should flip, which is easy to see on the 2x2 above, where swapping the rows gives rows -5, 4 and 3, -2, and a determinant of 10 - 12 = -2 instead of 2. Finally, look for a row or column of zeros: if any line is all zeros, the determinant is zero and no expansion is needed.

Decimals deserve one word of caution. The calculator snaps any result within one billionth of zero to exactly zero and rounds the display to six decimal places, because binary floating point cannot store 0.1 exactly and a determinant that should be zero can arrive as a tiny residue. Integer and simple decimal entries, which is most homework, come out exact.