How do I find the determinant of a 2×2 matrix?

Type the four numbers into grid A and read the large number to the right of the equals sign. The rule is ad − bc, the product of the main diagonal less the product of the other one. The page opens on [1 2; 3 4], where that is 1×4 − 2×3, so the answer on screen is -2. Put [4 7; 2 6] in instead and it is 24 − 14, or 10. A cell takes a whole number, a decimal like 0.5, a fraction like 1/3, exponent notation like 1.5e1, or the long minus sign that comes with anything pasted out of a PDF or a textbook, and a cell left blank counts as zero.

How does it work out a 3×3 or an 8×8 determinant?

The calculator uses exact Bareiss elimination, with row swaps accounted for in the sign. The row and column controls support 1×1 through 8×8. [6 1 1; 4 -2 5; 2 8 7] gives -306; [2 -1 0 0; -1 2 -1 0; 0 -1 2 -1; 0 0 -1 2] gives 5. It displays the finished determinant rather than a cofactor expansion or intermediate elimination steps.

What does a determinant of 0 mean?

The square matrix is singular and has no ordinary inverse. Its rows are dependent; Rank of A reports their independent count. [1 2 3; 4 5 6; 7 8 9] has determinant 0 and rank 2, and [1 2; 1 2] has determinant 0 and rank 1. Exact arithmetic reaches zero without relying on a numerical tolerance.

Why is my determinant showing as a fraction?

Because it is one. Fractions in give a fraction out: [1/2 1/3; 1/4 1/5] gives 1/60, and the 3×3 Hilbert matrix [1 1/2 1/3; 1/2 1/3 1/4; 1/3 1/4 1/5] gives 1/2160 exactly. Decimals you type are converted exactly on the way in as well, so [0.5 1.5; 2 4] gives -1 rather than something ending in stray digits. When a decimal is what you need, the Display switch under the stage turns the same exact value into one at 0 to 10 places on the stepper: 1/2160 reads 0.0005 at four places and 0.000462963 at ten, while the value behind it is still exactly 1/2160.

Can I take the determinant of a rectangular matrix?

An ordinary determinant is defined only for a square matrix. A 2×3, 3×4 or 4×3 matrix is refused with its exact shape and a suggestion to give A equal rows and columns. Rank and RREF still work for rectangles: [1 2 3; 4 5 6] has rank 2 and RREF [1 0 -1; 0 1 2]. This explanation does not claim to define a rectangular determinant.

Does swapping or scaling a row change the determinant?

Yes, in fixed ways, and retyping a matrix here is the quickest way to watch them. Swapping two rows flips the sign: [6 1 1; 4 -2 5; 2 8 7] is -306, and with its last two rows exchanged it is 306. Multiplying one row by k multiplies the determinant by k: double the first row of that same matrix and it becomes -612. Adding a multiple of one row to another leaves the determinant alone, which is exactly why elimination can be used to compute it. A triangular matrix is the easy case, since the answer is the product of its diagonal: [2 0 0; 5 3 0; 9 8 4] is 2 × 3 × 4, or 24.

Does opening this page change the calculator I use for other operations?

The page selects its stated operation after saved restoration. Simply visiting it does not overwrite the saved draft. Only fields you deliberately edit update their stored values; changing a display preference does not save a preset matrix. Everything is computed in your browser.

Matrix Determinant Calculator

Find the determinant of a square real matrix from 1×1 to 8×8 in exact fractions. This page selects det A; the opening [1 2; 3 4] gives -2 unless a saved matrix restores. Fill grid A or apply pasted rows. [1/2 1/3; 1/4 1/5] gives 1/60, and [1 2 3; 4 5 6; 7 8 9] gives exactly 0 with rank 2. Rectangular matrices have no ordinary determinant; the page explains their shape rather than inventing a value. Copy the result as text or LaTeX.

A2 × 2

RowsColumns

-2

Display

The scale factor of the matrix. Zero means the matrix is singular and has no inverse.

Paste a whole matrix

Use one row per line or semicolon. Separate entries with spaces, tabs or commas; write fractions as 1/3. Brackets around each row are optional. Applying a matrix replaces that grid and its dimensions.

Matrix A text

Determinantnot zero, so A can be inverted
-2
Rank of Aall 2 rows independent; the most a 2×2 matrix can have
2
det Aexact fractions
-2

Computed by Bareiss elimination in exact rational arithmetic, so nothing here is rounded.

Every calculation uses exact rational numbers. Integers, decimals and fractions are read exactly. A singular square matrix has determinant 0 and no ordinary inverse. Fraction display keeps the exact value; Decimal rounds only the displayed result. Floating-point methods need care with rounding residuals near zero. This calculator accepts real numeric entries, not complex numbers or symbolic variables. It shows the finished answer rather than an intermediate row-operation trace.

Common questions

How do I find the determinant of a 2×2 matrix?: Type the four numbers into grid A and read the large number to the right of the equals sign. The rule is ad − bc, the product of the main diagonal less the product of the other one. The page opens on [1 2; 3 4], where that is 1×4 − 2×3, so the answer on screen is -2. Put [4 7; 2 6] in instead and it is 24 − 14, or 10. A cell takes a whole number, a decimal like 0.5, a fraction like 1/3, exponent notation like 1.5e1, or the long minus sign that comes with anything pasted out of a PDF or a textbook, and a cell left blank counts as zero.
How does it work out a 3×3 or an 8×8 determinant?: The calculator uses exact Bareiss elimination, with row swaps accounted for in the sign. The row and column controls support 1×1 through 8×8. [6 1 1; 4 -2 5; 2 8 7] gives -306; [2 -1 0 0; -1 2 -1 0; 0 -1 2 -1; 0 0 -1 2] gives 5. It displays the finished determinant rather than a cofactor expansion or intermediate elimination steps.
What does a determinant of 0 mean?: The square matrix is singular and has no ordinary inverse. Its rows are dependent; Rank of A reports their independent count. [1 2 3; 4 5 6; 7 8 9] has determinant 0 and rank 2, and [1 2; 1 2] has determinant 0 and rank 1. Exact arithmetic reaches zero without relying on a numerical tolerance.
Why is my determinant showing as a fraction?: Because it is one. Fractions in give a fraction out: [1/2 1/3; 1/4 1/5] gives 1/60, and the 3×3 Hilbert matrix [1 1/2 1/3; 1/2 1/3 1/4; 1/3 1/4 1/5] gives 1/2160 exactly. Decimals you type are converted exactly on the way in as well, so [0.5 1.5; 2 4] gives -1 rather than something ending in stray digits. When a decimal is what you need, the Display switch under the stage turns the same exact value into one at 0 to 10 places on the stepper: 1/2160 reads 0.0005 at four places and 0.000462963 at ten, while the value behind it is still exactly 1/2160.
Can I take the determinant of a rectangular matrix?: An ordinary determinant is defined only for a square matrix. A 2×3, 3×4 or 4×3 matrix is refused with its exact shape and a suggestion to give A equal rows and columns. Rank and RREF still work for rectangles: [1 2 3; 4 5 6] has rank 2 and RREF [1 0 -1; 0 1 2]. This explanation does not claim to define a rectangular determinant.
Does swapping or scaling a row change the determinant?: Yes, in fixed ways, and retyping a matrix here is the quickest way to watch them. Swapping two rows flips the sign: [6 1 1; 4 -2 5; 2 8 7] is -306, and with its last two rows exchanged it is 306. Multiplying one row by k multiplies the determinant by k: double the first row of that same matrix and it becomes -612. Adding a multiple of one row to another leaves the determinant alone, which is exactly why elimination can be used to compute it. A triangular matrix is the easy case, since the answer is the product of its diagonal: [2 0 0; 5 3 0; 9 8 4] is 2 × 3 × 4, or 24.
Does opening this page change the calculator I use for other operations?: The page selects its stated operation after saved restoration. Simply visiting it does not overwrite the saved draft. Only fields you deliberately edit update their stored values; changing a display preference does not save a preset matrix. Everything is computed in your browser.

Exact: matrices are computed in rational arithmetic, so an inverse of an integer matrix is shown as fractions rather than rounded decimals, and a singular matrix is reported as singular instead of producing huge numbers.