Determinant Calculator
This determinant calculator works out the determinant of a 2×2, 3×3 or 4×4 matrix, writes the whole expansion out, and keeps everything in exact fractions — so a determinant of zero is really zero, not a rounded-off guess at it.
Start small. For 1 2 / 3 4, one diagonal minus the other gives (1)(4) − (2)(3) = −2. Now the standard 3×3, 6 1 1 / 4 −2 5 / 2 8 7: expanding along the top row gives 6[−54] − 1[18] + 1[36] = −306. The three numbers in brackets are smaller determinants, and the next sections are about where they come from.
You will see the matrix determinant written both ways: det(A), or |A| with straight bars in place of the usual brackets. Either notation means the same thing — one number, a scalar, squeezed out of a whole square matrix. That is what makes it so useful: a single value that answers several questions about the matrix at once.
Finding the Determinant of a 2×2 Matrix
Top-left times bottom-right, minus top-right times bottom-left. That is the entire rule for a 2x2, and everything larger is built out of it.
It is worth knowing what the number is measuring. A matrix stretches and shears the plane, and det(A) is the factor by which it scales area. A value of 3 means shapes come out three times bigger; −2 means twice as big and flipped over, the way a mirror flips a hand. And a value of nothing at all means the plane has been squashed onto a line, which is the whole story of the zero case further down.
Finding the Determinant of a 3×3 Matrix
Take each entry along the top row, cross out its own row and column, and take the determinant of the 2×2 that is left. Multiply, and alternate the signs: plus, minus, plus. This is cofactor expansion — also called Laplace expansion — and the small determinants it leans on are the minors.
With 6 1 1 / 4 −2 5 / 2 8 7, crossing out the top row and the first column leaves −2 5 / 8 7, whose determinant is (−2)(7) − (5)(8) = −54. The second leaves 4 5 / 2 7 = 18, and the third leaves 4 −2 / 2 8 = 36. So the answer is 6(−54) − 1(18) + 1(36) = −306.
There is a second way, and it works only at this size. The rule of Sarrus adds the three products running down to the right and subtracts the three running down to the left: (6)(−2)(7) + (1)(5)(2) + (1)(4)(8) − (6)(5)(8) − (1)(4)(7) − (1)(−2)(2), which comes to −306 as well. This calculator runs both and compares them before it shows you anything — two routes, one answer, which is as close to a proof as a calculator can offer.
Determinants of Bigger Matrices
A 4×4 works exactly the same way: four entries along the top row, each multiplied by the 3×3 left behind, with the signs alternating. The only thing that changes is the workload. Each of those four 3×3s costs three 2×2s, so one 4x4 determinant is twelve minors underneath — which is why this page stops there. The method still holds at 5×5, but 120 signed products is not a hand calculation, and pretending otherwise helps nobody.
One property is worth carrying around because it saves real work: the determinant of a product is the product of the determinants. If you already know det(A) and det(B), there is no need to multiply the matrices out first — and if you want to see that for yourself, the matrix multiplication calculator will give you AB to check it against.
Some matrices give the answer away. If everything on one side of the diagonal is zero, the determinant is just the diagonal multiplied together: 2 0 0 / 0 3 0 / 0 0 4 is 24, and no expansion is needed. The calculator notices that and says so.
What a Determinant of Zero Means
A value of zero means the rows are linearly dependent — one of them is built out of the others — and the matrix is called singular. Anything else, however small, means the matrix is invertible, or non-singular. Take 1 1 1 / 2 5 −1 / 2 5 −1: the last two rows are identical, so the answer is zero before any arithmetic is done. The four-by-four of 1 to 16, row by row, is zero for the same kind of reason.
In the picture from earlier, this is the transformation collapsing space onto something flatter: an area scaled by zero. Nothing that has been flattened can be unflattened, which is why a singular matrix cannot be undone and has no inverse, and why a system of equations built on one has no single answer. That is the exact point where Cramer’s rule runs out of road: it divides by this number, and it cannot divide by nothing.
This is also the exact place where a calculator that rounds can mislead you, and at least one well-ranked tool of this kind says so in its own documentation: rounding can hide a genuinely singular matrix when the value lands just above or below zero. Consider 0.1 0.2 0.3 / 0.2 0.4 0.6 / 1 5 9, where the second row is twice the first. Its value is exactly zero. Now consider 1 1 / 1 1.0000000001, which is not singular at all — det(A) there is one ten-billionth. A tool working in rounded decimals can easily report the first as a tiny non-zero number and the second as zero, getting both backwards. Everything here is done in fractions, so that never happens.
How to Use This Determinant Calculator
The screen has three parts: the answer band at the top, the matrix you fill in below it, and the keypad underneath. You fill cells, the way you would write the matrix out by hand.
1. Set the size. Two rows of number buttons sit above the grid, one for rows and one for columns, from 1 to 4. They are separate on purpose — make it 2×3 and the calculator will tell you there is no determinant to find, which is a question worth asking once.
2. Fill in the cells. Tap a cell to select it, then use the keypad. A cell takes a whole number, a decimal such as 0.25, or a fraction such as 3/4, and anything left blank counts as zero.
3. Press =. The determinant appears at the top. If it is zero, the caption changes to SINGULAR and the line underneath tells you what that means for this matrix.
4. Open the expansion. The button marked Show the expansion lays out the top row term by term, then each minor as its own block — the smaller matrix left after crossing out, and its own arithmetic — and finally the Sarrus check on a 3×3.
Every key and control, in one place:
| rows / columns | Sets the size, 1 to 4, each one on its own |
| the cells | Tap to select; blank counts as zero; fractions and decimals both accepted |
| ← → | Move to the previous or next cell, wrapping round the whole grid |
| ↑ ↓ | Move up or down a row, for filling a column at a time |
| ± | Flips the sign of the cell you are in, so a minus cannot be typed twice by accident |
| ⁄ | The fraction bar. 3⁄4 stays a fraction all the way to the answer |
| . | Decimal point. 0.25 is read as one quarter, exactly |
| ⌫ | Deletes one character from the cell you are in |
| AC | Clears the whole matrix and starts again |
| = | Works out the determinant |
| My keyboard | Switches that cell to your own keyboard, for anything the pad does not carry |
| copy | The icon in the band copies the answer |
Change anything after an answer is showing and the band dims and asks you to press = again, so what you are reading always belongs to what is in the cells. There are no limits, no ads and no account.
Determinant FAQ
How do you find the determinant of a 2x2 matrix?
What is a singular matrix?
Can a non-square matrix have a determinant?
What is the rule of Sarrus?
How do you find the determinant of a 4x4 matrix?
What does a negative determinant mean?
One diagonal minus the other, then the same idea nested inside itself for anything bigger. The number that falls out tells you how much the matrix stretches space — and if it is zero, that the matrix has flattened space altogether.