Determinant Calculator (2×2, 3×3, 4×4)

Choose the matrix size (2×2 to 4×4) and enter the entries in the grid. A determinant is defined only for a square matrix (rows = columns).

Matrix size

Enter the entries as numbers (decimals and negatives are fine).

Result

Choose the matrix size on the left, enter the entries and press "Calculate". The determinant and the steps will appear here.

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What you can do on this page
What is this calculation used for?
How to Use
Formula
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Find the determinant \(\det(A)\) of a 2×2, 3×3 or 4×4 square matrix on the spot
Steps follow the textbook methods: "\(ad - bc\)" for 2×2, the rule of Sarrus for 3×3 and cofactor expansion for 4×4
It also tells you whether the matrix is invertible: if \(\det(A) \neq 0\), an inverse exists
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets (MDETERM) and Python are all on this page

A determinant is defined only for a square matrix (rows = columns). Matrix addition, multiplication, transpose and more are on the sister page "Matrix Calculator".

What is this calculation used for?

Checking whether a system of equations has one solution (Cramer's rule)

For a system of linear equations with as many equations as unknowns, if the determinant of the square matrix of coefficients is not 0, the system has exactly one solution. If the determinant is 0, there is either no solution or infinitely many.
With Cramer's rule, you can even write the solution itself using only ratios of determinants. This is part of the theory behind engineering work that solves huge numbers of equations, such as structural analysis and circuit analysis.

Showing how many times a shape's area or volume grows (scale factor of a linear transformation)

When you transform a flat shape with a 2×2 matrix (rotate it, stretch or shrink it, or shear it), its area is multiplied by exactly \(|\det(A)|\). For a solid and a 3×3 matrix, the volume is multiplied by \(|\det(A)|\).
This role of the determinant as a scale factor also appears in college calculus as the correction factor for a change of variables in an integral (the Jacobian), and it is used all over science and engineering.

Detecting "flipped" shapes in 3D graphics and games

3D graphics software and game engines use matrices to transform models. If the determinant of a transformation matrix is negative, the transformation flips the model like a mirror image, and the front and back of each face (the direction of its normal) are swapped.
So graphics software checks the sign of the determinant to detect flipping and corrects it so that the drawing does not break.

Finding area from coordinates (surveying and maps)

If you know the coordinates of a triangle's vertices, its area is half the absolute value of a determinant built from those coordinates. Extended to polygons, this is called the shoelace formula, a standard way to calculate land area in surveying and in mapping (GIS) software.
The secret behind "enter the coordinates and get the area" is exactly the determinant.

The doorway to eigenvalues (vibration analysis and data analysis)

Vibration analysis, which studies how a building or machine tends to shake, and principal component analysis, which pulls the main trends out of large data sets, both use values of a matrix called eigenvalues.
Eigenvalues are found from the equation "find \(\lambda\) such that \(\det(A - \lambda I) = 0\)" (the characteristic equation). The determinant is also a doorway to this more advanced tool.

Formula

2×2 determinant (ad − bc)

Standard notation (the usual math form)

\(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\) \(=\) \(ad\) \(-\) \(bc\)

In words (symbols replaced with words)

③ \(\det(A)\): 2×2 determinant \(=\) ① \(ad\): top left × bottom right \(-\) ② \(bc\): top right × bottom left

The formula in words

① From the \(ad\): product of top left × bottom right ,

② subtract the \(bc\): product of top right × bottom left , and you get the

③ \(\det(A)\): 2×2 determinant

Quick example

The determinant of the matrix \(\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}\) is

\(\det(A)\): determinant \(=\) top left × bottom right (\(4 \times 6\)) \(-\) top right × bottom left (\(7 \times 2\))

\(\begin{vmatrix} 4 & 7 \\ 2 & 6 \end{vmatrix} = 4 \times 6 - 7 \times 2 = 24 - 14 = 10\)

Key idea

The diagonal from top left to bottom right gets "+", and the diagonal from top right to bottom left gets "−". You just multiply along the two diagonals and subtract, so this criss-cross pattern is easy to remember. The determinant is written \(|A|\) as well as \(\det(A)\) (the bars look like absolute value, but here they have a different job). A matrix with \(\det(A) = 0\) is called "not invertible" (singular), and it has no inverse.

3×3 determinant (rule of Sarrus)

Standard notation (the usual math form)

\(\det(A)\) \(=\) \(aei + bfg + cdh\) \(-\) \((ceg + afh + bdi)\)

In words (symbols replaced with words)

③ \(\det(A)\): 3×3 determinant \(=\) ① sum of the products on the 3 down-right diagonals \(-\) ② sum of the products on the 3 up-right diagonals

The formula in words

① With the entries named \(A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\), take the sum of the products on the 3 down-right diagonals (\(aei + bfg + cdh\)) ,

② subtract the sum of the products on the 3 up-right diagonals (\(ceg + afh + bdi\)) , and you get the

③ \(\det(A)\): 3×3 determinant

Quick example

The determinant of \(\begin{pmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 1 & 1 & 1 \end{pmatrix}\) is

\(1 \times 3 \times 1 + 0 \times 1 \times 1 + 2 \times (-1) \times 1 = 3 + 0 - 2 = 1\)

\(2 \times 3 \times 1 + 1 \times 1 \times 1 + 0 \times (-1) \times 1 = 6 + 1 + 0 = 7\)

\(\begin{vmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 1 & 1 & 1 \end{vmatrix} = 1 - 7 = -6\)

Key idea

The rule of Sarrus is easier to use if you copy columns 1 and 2 again to the right of the matrix. There are 3 diagonals from top left to bottom right (\(aei\), \(bfg\), \(cdh\), all "+") and 3 diagonals from top right to bottom left (\(ceg\), \(afh\), \(bdi\), all "−"), 6 criss-cross products in all. However, the rule of Sarrus works only up to 3×3 (the 2×2 formula \(ad-bc\) uses the same criss-cross idea). It does not work for 4×4 or larger, so use the cofactor expansion below instead.

Cofactor expansion (the general method, also for 4×4 and larger)

Standard notation (the usual math form)

\(\det(A)\) \(=\) \(\sum_{j=1}^{n}\) \((-1)^{1+j}\) \(a_{1j}\) \(M_{1j}\)

In words (symbols replaced with words)

⑤ \(\det(A)\): \(n\)×\(n\) determinant \(=\) ④ add up for \(j = 1\) to \(n\) ③ \((-1)^{1+j}\): sign (alternating + − + − …) ① \(a_{1j}\): entry in row 1, column \(j\) ② \(M_{1j}\): minor without row 1 and column \(j\)

The formula in words

① Multiply the \(a_{1j}\): entry in row 1, column \(j\)

② by the \(M_{1j}\): minor, the determinant left after removing row 1 and column \(j\) ,

③ attach the \((-1)^{1+j}\): sign (alternating + − + − …) ,

④ and add them up for \(j = 1\) to \(n\) to get the

⑤ \(\det(A)\): \(n\)×\(n\) determinant

Quick example

Expanding the 3×3 example above along row 1 gives (the same answer as the rule of Sarrus)

\(\begin{vmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 1 & 1 & 1 \end{vmatrix} = 1 \times \begin{vmatrix} 3 & 1 \\ 1 & 1 \end{vmatrix} - 0 \times \begin{vmatrix} -1 & 1 \\ 1 & 1 \end{vmatrix} + 2 \times \begin{vmatrix} -1 & 3 \\ 1 & 1 \end{vmatrix}\)

\(= 1 \times 2 - 0 \times (-2) + 2 \times (-4) = 2 - 0 - 8 = -6\)

Key idea

Cofactor expansion (also called Laplace expansion) breaks a large determinant into a combination of determinants one size smaller (minors). A 4×4 becomes four 3×3 determinants, and each 3×3 can be found with the rule of Sarrus (or by expanding again). This calculator shows the steps for 4×4 matrices this way. You do not have to expand along row 1; any row or any column gives the same answer. Choosing a row or column with many 0s makes those terms vanish and the work easier. The signs \((-1)^{i+j}\) alternate + and − like a checkerboard, starting with "+" at the top left.

A determinant is "a single number determined by a square matrix". Use \(ad - bc\) for 2×2, the rule of Sarrus for 3×3 and cofactor expansion for 4×4 and larger. Whether \(\det(A) \neq 0\) tells you if an inverse exists (that is, if the system of equations has exactly one solution).

Symbols and terms

Symbols

\(A\)	A	The name given to a matrix. By custom, matrices are named with capital letters.
\(\det(A),\ \|A\|\)	determinant of A	The determinant of matrix \(A\). The vertical-bar form \(\|A\|\) is also common (it is the same sign as absolute value for numbers, but around a matrix it stands for the determinant).
\(a_{ij}\)	a sub i j	The entry of matrix \(A\) in row \(i\), column \(j\) (the number in the \(i\)th row from the top and the \(j\)th column from the left). The small letters at the lower right (the subscripts) give its position.
\(M_{ij}\)	M sub i j	A minor: the determinant of the one-size-smaller matrix left after removing row \(i\) and column \(j\) from matrix \(A\).
\((-1)^{i+j}\)	negative 1 to the power i plus j	The sign used in cofactor expansion. It is \(+1\) when \(i+j\) is even and \(-1\) when it is odd, so + and − alternate like a checkerboard, starting with "+" at the top left.
\(\sum\)	sigma	The sign for "add them all up". \(\sum_{j=1}^{n}\) tells you to add up the terms for every \(j\) from 1 to \(n\).

Terms

determinant	A single number determined by a square matrix. It packs key facts about the matrix (whether it has an inverse, how many times a transformation scales area or volume, and so on) into one number.
square matrix	A matrix with the same number of rows and columns (shaped like a square). Determinants and inverses exist only for square matrices.
rule of Sarrus	A way to find a 3×3 determinant by adding and subtracting the products along 6 diagonals. It is also called the diagonal method. It is named after the French mathematician Pierre Sarrus and works only for 3×3 matrices.
cofactor expansion	A method that splits a determinant into a sum of "entry × minor × sign", turning it into determinants one size smaller. It is also called Laplace expansion, and it works for 4×4 and larger determinants.
minor	The determinant of the smaller matrix left after removing one row and one column. It is a building block of cofactor expansion.
cofactor	A minor \(M_{ij}\) multiplied by the sign \((-1)^{i+j}\). "Cofactor expansion" gets its name from expanding into a sum of these cofactors.
invertible matrix	A square matrix whose determinant is not 0. It is also called nonsingular, and it has an inverse. A matrix with determinant 0 is called singular and has no inverse.
inverse matrix	The matrix that gives the identity matrix (1s on the diagonal and 0s elsewhere) when multiplied. It is the matrix version of a reciprocal and is written \(A^{-1}\). It exists only when the determinant is not 0.
linear algebra	The branch of mathematics that deals with matrices and vectors. It is often a required first- or second-year college course and is the foundation of computer graphics, statistics, AI and more.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.

Operations with negative numbers (Grade 7)	Being able to multiply and subtract with negative numbers without sign mistakes
Variables and subscripts (Grade 6 to high school)	Being able to read the small characters at the lower right, as in \(a_{12}\) (subscripts), as labels for a position, not as values
Powers and signs (Grades 6–7)	Knowing that \((-1)^{n}\) alternates: \(+1\) when \(n\) is even and \(-1\) when \(n\) is odd
Matrix basics (linear algebra)	Knowing the words row (horizontal line), column (vertical line), entry and square matrix (explained on the sister page "Matrix Calculator")

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find a 2×2 determinant

Row 1 of matrix A	4	7
Row 2 of matrix A	2	6
Determinant det(A)	=MDETERM(B1:C2)

Table to find a 3×3 determinant

Row 1 of matrix A	1	0	2
Row 2 of matrix A	-1	3	1
Row 3 of matrix A	1	1	1
Determinant det(A)	=MDETERM(B1:D3)

Table to find a 4×4 determinant

Row 1 of matrix A	1	2	0	1
Row 2 of matrix A	0	1	3	2
Row 3 of matrix A	2	0	1	1
Row 4 of matrix A	1	1	0	2
Determinant det(A)	=MDETERM(B1:E4)

The MDETERM function alone finds the determinant. Just give it the range of the square matrix (B1:C2 in the first table), and the answer goes into a single cell.
The first table gives 10, the second gives -6 and the third gives 18.
When MDETERM returns 0, the matrix has no inverse (the MINVERSE function, which finds the inverse, returns an error).

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find a 2×2 determinant

Row 1 of matrix A	4	7
Row 2 of matrix A	2	6
Determinant det(A)	=MDETERM(B1:C2)

Table to find a 3×3 determinant

Row 1 of matrix A	1	0	2
Row 2 of matrix A	-1	3	1
Row 3 of matrix A	1	1	1
Determinant det(A)	=MDETERM(B1:D3)

The same MDETERM function as in Excel works as is. Copy the whole table and paste it into cell A1.
The first table gives 10 and the second gives -6.

How to calculate it in Python

A = [[1, 0, 2], [-1, 3, 1], [1, 1, 1]]

def determinant(matrix):    # cofactor expansion (computed recursively along row 1)
    n = len(matrix)
    if n == 1:
        return matrix[0][0]
    total = 0
    for j in range(n):
        minor = [row[:j] + row[j + 1:] for row in matrix[1:]]   # submatrix without row 1 and column j
        total += (-1) ** j * matrix[0][j] * determinant(minor)
    return total

print("det(A) =", determinant(A))

Runs with the standard library only. Replace A at the top with your own matrix (any square matrix, even 4×4 or larger) and run it. This example gives det(A) = -6. For serious numerical work, the standard choice is NumPy, a library made for matrices (numpy.linalg.det(A)).

How to write it in LaTeX and other math languages (copy and paste)

2×2 determinant (ad − bc)

|A| = ad − bc

\det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>det</mi><mo>&#x2061;</mo><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow>
    <mo>=</mo>
    <mi>a</mi><mi>d</mi>
    <mo>-</mo>
    <mi>b</mi><mi>c</mi>
  </mrow>
</math>

det(A) = |(a, b), (c, d)| = ad - bc

Det[{{a, b}, {c, d}}]

LinearAlgebra:-Determinant(Matrix([[a, b], [c, d]]));

d = det(A);

det(A) = ad − bc

3×3 determinant (rule of Sarrus)

|A| = aei + bfg + cdh − ceg − afh − bdi

\det(A) = aei + bfg + cdh - ceg - afh - bdi

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>det</mi><mo>&#x2061;</mo><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow>
    <mo>=</mo>
    <mi>a</mi><mi>e</mi><mi>i</mi>
    <mo>+</mo>
    <mi>b</mi><mi>f</mi><mi>g</mi>
    <mo>+</mo>
    <mi>c</mi><mi>d</mi><mi>h</mi>
    <mo>-</mo>
    <mi>c</mi><mi>e</mi><mi>g</mi>
    <mo>-</mo>
    <mi>a</mi><mi>f</mi><mi>h</mi>
    <mo>-</mo>
    <mi>b</mi><mi>d</mi><mi>i</mi>
  </mrow>
</math>

det(A) = aei + bfg + cdh - ceg - afh - bdi

Det[{{a, b, c}, {d, e, f}, {g, h, i}}]

LinearAlgebra:-Determinant(Matrix([[a, b, c], [d, e, f], [g, h, i]]));

d = det(A);

det(A) = aei + bfg + cdh − ceg − afh − bdi

Cofactor expansion (the general method, also for 4×4 and larger)

det(A) = Σ (−1)¹⁺ʲ a₁ⱼ M₁ⱼ  (j = 1 to n)

\det(A) = \sum_{j=1}^{n} (-1)^{1+j} a_{1j} M_{1j}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>det</mi><mo>&#x2061;</mo><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow>
    <mo>=</mo>
    <munderover>
      <mo>&#x2211;</mo>
      <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow>
      <mi>n</mi>
    </munderover>
    <msup>
      <mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow>
      <mrow><mn>1</mn><mo>+</mo><mi>j</mi></mrow>
    </msup>
    <msub><mi>a</mi><mrow><mn>1</mn><mi>j</mi></mrow></msub>
    <msub><mi>M</mi><mrow><mn>1</mn><mi>j</mi></mrow></msub>
  </mrow>
</math>

det(A) = sum_(j=1)^(n) (-1)^(1+j) a_(1j) M_(1j)

Det[A]

LinearAlgebra:-Determinant(A);

d = det(A);

det(A) = ∑_(j=1)^n (−1)^(1+j) a_1j M_1j

How to have ChatGPT do the calculation

You are a calculation assistant for matrices. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

Find the determinant (det) of each of the following two square matrices.
1. A = [[1, 0, 2], [-1, 3, 1], [1, 1, 1]]
2. B = [[1, 2], [2, 4]]

For each one, show the value of the determinant and whether the matrix is invertible (whether the determinant is nonzero, that is, whether an inverse exists).

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula