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Inverse Matrix Calculator

Find A⁻¹ for a square matrix, or learn why it does not exist.

Matrix A
× same

Enter integers, decimals or fractions such as 3/4. Blank cells count as 0.

Show decimals instead of fractions, in the steps too

Inverse

A−1=[1319−119−719−419−719819−119319219]A^{-1}=\begin{bmatrix} \frac{13}{19} & -\frac{1}{19} & -\frac{7}{19} \\ -\frac{4}{19} & -\frac{7}{19} & \frac{8}{19} \\ -\frac{1}{19} & \frac{3}{19} & \frac{2}{19} \end{bmatrix}A−1=​1913​−194​−191​​−191​−197​193​​−197​198​192​​​

Steps

  1. 1Determinant
    det⁡A\det AdetA
    ↓
    det⁡A=−19\det A=-19detA=−19

    Non-zero, so the inverse exists.

  2. 2Augment with the identity
    [A∣I][A\mid I][A∣I]
    ↓
    [2131000−14010125001]\left[\begin{array}{ccc|ccc} 2 & 1 & 3 & 1 & 0 & 0 \\ 0 & -1 & 4 & 0 & 1 & 0 \\ 1 & 2 & 5 & 0 & 0 & 1 \end{array}\right]​201​1−12​345​100​010​001​​

    Row-reduce until the left half is the identity; the right half is then A⁻¹.

  3. 3R1 ← R1 ÷ 2
    [2131000−14010125001]\left[\begin{array}{ccc|ccc} 2 & 1 & 3 & 1 & 0 & 0 \\ 0 & -1 & 4 & 0 & 1 & 0 \\ 1 & 2 & 5 & 0 & 0 & 1 \end{array}\right]​201​1−12​345​100​010​001​​
    ↓
    [1123212000−14010125001]\left[\begin{array}{ccc|ccc} 1 & \frac{1}{2} & \frac{3}{2} & \frac{1}{2} & 0 & 0 \\ 0 & -1 & 4 & 0 & 1 & 0 \\ 1 & 2 & 5 & 0 & 0 & 1 \end{array}\right]​101​21​−12​23​45​21​00​010​001​​
  4. 4R3 ← R3 − 1·R1
    [1123212000−14010125001]\left[\begin{array}{ccc|ccc} 1 & \frac{1}{2} & \frac{3}{2} & \frac{1}{2} & 0 & 0 \\ 0 & -1 & 4 & 0 & 1 & 0 \\ 1 & 2 & 5 & 0 & 0 & 1 \end{array}\right]​101​21​−12​23​45​21​00​010​001​​
    ↓
    [1123212000−1401003272−1201]\left[\begin{array}{ccc|ccc} 1 & \frac{1}{2} & \frac{3}{2} & \frac{1}{2} & 0 & 0 \\ 0 & -1 & 4 & 0 & 1 & 0 \\ 0 & \frac{3}{2} & \frac{7}{2} & -\frac{1}{2} & 0 & 1 \end{array}\right]​100​21​−123​​23​427​​21​0−21​​010​001​​
  5. 5R2 ← R2 ÷ -1
    [1123212000−1401003272−1201]\left[\begin{array}{ccc|ccc} 1 & \frac{1}{2} & \frac{3}{2} & \frac{1}{2} & 0 & 0 \\ 0 & -1 & 4 & 0 & 1 & 0 \\ 0 & \frac{3}{2} & \frac{7}{2} & -\frac{1}{2} & 0 & 1 \end{array}\right]​100​21​−123​​23​427​​21​0−21​​010​001​​
    ↓
    [11232120001−40−1003272−1201]\left[\begin{array}{ccc|ccc} 1 & \frac{1}{2} & \frac{3}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 1 & -4 & 0 & -1 & 0 \\ 0 & \frac{3}{2} & \frac{7}{2} & -\frac{1}{2} & 0 & 1 \end{array}\right]​100​21​123​​23​−427​​21​0−21​​0−10​001​​
  6. 6R1 ← R1 − 1/2·R2
    [11232120001−40−1003272−1201]\left[\begin{array}{ccc|ccc} 1 & \frac{1}{2} & \frac{3}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 1 & -4 & 0 & -1 & 0 \\ 0 & \frac{3}{2} & \frac{7}{2} & -\frac{1}{2} & 0 & 1 \end{array}\right]​100​21​123​​23​−427​​21​0−21​​0−10​001​​
    ↓
    [10721212001−40−1003272−1201]\left[\begin{array}{ccc|ccc} 1 & 0 & \frac{7}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 1 & -4 & 0 & -1 & 0 \\ 0 & \frac{3}{2} & \frac{7}{2} & -\frac{1}{2} & 0 & 1 \end{array}\right]​100​0123​​27​−427​​21​0−21​​21​−10​001​​
  7. 7R3 ← R3 − 3/2·R2
    [10721212001−40−1003272−1201]\left[\begin{array}{ccc|ccc} 1 & 0 & \frac{7}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 1 & -4 & 0 & -1 & 0 \\ 0 & \frac{3}{2} & \frac{7}{2} & -\frac{1}{2} & 0 & 1 \end{array}\right]​100​0123​​27​−427​​21​0−21​​21​−10​001​​
    ↓
    [10721212001−40−1000192−12321]\left[\begin{array}{ccc|ccc} 1 & 0 & \frac{7}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 1 & -4 & 0 & -1 & 0 \\ 0 & 0 & \frac{19}{2} & -\frac{1}{2} & \frac{3}{2} & 1 \end{array}\right]​100​010​27​−4219​​21​0−21​​21​−123​​001​​
  8. 8R3 ← R3 ÷ 19/2
    [10721212001−40−1000192−12321]\left[\begin{array}{ccc|ccc} 1 & 0 & \frac{7}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 1 & -4 & 0 & -1 & 0 \\ 0 & 0 & \frac{19}{2} & -\frac{1}{2} & \frac{3}{2} & 1 \end{array}\right]​100​010​27​−4219​​21​0−21​​21​−123​​001​​
    ↓
    [10721212001−40−10001−119319219]\left[\begin{array}{ccc|ccc} 1 & 0 & \frac{7}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 1 & -4 & 0 & -1 & 0 \\ 0 & 0 & 1 & -\frac{1}{19} & \frac{3}{19} & \frac{2}{19} \end{array}\right]​100​010​27​−41​21​0−191​​21​−1193​​00192​​​
  9. 9R1 ← R1 − 7/2·R3
    [10721212001−40−10001−119319219]\left[\begin{array}{ccc|ccc} 1 & 0 & \frac{7}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 1 & -4 & 0 & -1 & 0 \\ 0 & 0 & 1 & -\frac{1}{19} & \frac{3}{19} & \frac{2}{19} \end{array}\right]​100​010​27​−41​21​0−191​​21​−1193​​00192​​​
    ↓
    [1001319−119−71901−40−10001−119319219]\left[\begin{array}{ccc|ccc} 1 & 0 & 0 & \frac{13}{19} & -\frac{1}{19} & -\frac{7}{19} \\ 0 & 1 & -4 & 0 & -1 & 0 \\ 0 & 0 & 1 & -\frac{1}{19} & \frac{3}{19} & \frac{2}{19} \end{array}\right]​100​010​0−41​1913​0−191​​−191​−1193​​−197​0192​​​
  10. 10R2 ← R2 + 4·R3
    [1001319−119−71901−40−10001−119319219]\left[\begin{array}{ccc|ccc} 1 & 0 & 0 & \frac{13}{19} & -\frac{1}{19} & -\frac{7}{19} \\ 0 & 1 & -4 & 0 & -1 & 0 \\ 0 & 0 & 1 & -\frac{1}{19} & \frac{3}{19} & \frac{2}{19} \end{array}\right]​100​010​0−41​1913​0−191​​−191​−1193​​−197​0192​​​
    ↓
    [1001319−119−719010−419−719819001−119319219]\left[\begin{array}{ccc|ccc} 1 & 0 & 0 & \frac{13}{19} & -\frac{1}{19} & -\frac{7}{19} \\ 0 & 1 & 0 & -\frac{4}{19} & -\frac{7}{19} & \frac{8}{19} \\ 0 & 0 & 1 & -\frac{1}{19} & \frac{3}{19} & \frac{2}{19} \end{array}\right]​100​010​001​1913​−194​−191​​−191​−197​193​​−197​198​192​​​
  1. Tools
  2. Mathematics
  3. Matrix Calculator
  4. Inverse Matrix Calculator

About Inverse Matrix Calculator

This calculator finds the inverse of a square matrix up to 6 × 6. It row-reduces the matrix next to the identity, [A | I], until the left side becomes I, and the right side is then A⁻¹. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

It starts by computing the determinant. If that is zero, the matrix is singular, and the calculator says there is no inverse instead of producing a wrong grid. For [[4, 7], [2, 6]] the determinant is 10, and the inverse is [[3/5, −7/10], [−1/5, 2/5]], with fractions kept exact. The decimal switch shows 0.6, −0.7, −0.2 and 0.4.

The defining property is A × A⁻¹ = I. Multiply your answer back as a check, using the multiplication page. The inverse is what lets you solve AX = B as X = A⁻¹B.

How to use Inverse Matrix Calculator

  1. 1Choose an operation, such as Determinant, Inverse or A × B.
  2. 2Set the size of matrix A with the row and column selectors. Square operations use a square grid, and two-matrix operations also show matrix B.
  3. 3Type the entries. Use integers, decimals or fractions like 3/4, and leave a cell blank for zero.
  4. 4The tool opens on a worked example. Read the result, typeset as a determinant, a rank or a matrix.
  5. 5Read the steps for the working, and use the decimal switch if you would rather see decimals than fractions.

Features

  • Grids up to 6 × 6, with rows and columns set separately for non-square matrices.
  • Addition, subtraction and multiplication of two matrices, with a clear message when the sizes do not match.
  • Determinant with the 2 × 2 rule, first-row expansion for 3 × 3 and row reduction for larger matrices.
  • Inverse by row-reducing [A | I], with a plain message when the matrix is singular.
  • Rank and reduced row echelon form with each row operation listed.
  • Adjugate and cofactor matrix for square matrices.
  • Eigenvalues from the characteristic polynomial, including complex ones, with eigenvectors: exact for rational eigenvalues and numeric (labelled) for irrational ones.
  • LU decomposition (with row swaps when a pivot is zero) and QR decomposition by Gram-Schmidt, exact with square roots.
  • Fraction entries and fraction results, with a switch to decimals that also applies to the steps.
  • Transpose of any matrix.

Inverse by Gauss-Jordan

Row-reduce the augmented matrix [A | I]. When the left block is I, the right block is the inverse. For 2 × 2 there is a direct formula.

[A | I]  →  [I | A⁻¹]
[[a, b], [c, d]]⁻¹ = (1 / (ad − bc)) × [[d, −b], [−c, a]]

Worked examples

  • [[4, 7], [2, 6]] → gives [[3/5, −7/10], [−1/5, 2/5]]

    det = 4×6 − 7×2 = 10. A⁻¹ = (1/10) × [[6, −7], [−2, 4]].

  • [[1, 2], [2, 4]] → gives No inverse

    The determinant is 0, so the matrix is singular.

Privacy: how it works

  • All the arithmetic runs in your browser tab. Your matrices are not sent to Ulkaa or anyone else, and there is nothing to sign in to.

Tips

  • Write fractions as 3/4 rather than 0.75 if you want results to stay exact.
  • Leave entries blank for zero instead of typing every zero in a large grid.
  • To solve a linear system, put the coefficients and constants in one grid and use RREF, or use the equation solver.
  • Multiplication is not commutative. To check AB against BA, swap which matrix is A and which is B.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Your matrices are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Your matrices are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

How do I find the inverse of a matrix?

Row-reduce [A | I] to [I | A⁻¹], or for a 2 × 2 use (1 ÷ det) times [[d, −b], [−c, a]].

When does an inverse not exist?

When the matrix is not square or its determinant is zero. The rows are then dependent.

More Matrix Calculator pages

  • Matrix CalculatorAdd, multiply, invert, decompose (LU, QR) and analyse matrices up to 6 × 6, with typeset working.
  • Matrix Determinant CalculatorFind the determinant of any square matrix up to 6 × 6.
  • Matrix Multiplication CalculatorMultiply matrix A by matrix B.
  • Matrix Addition CalculatorAdd or subtract two matrices of the same size.
  • Matrix Transpose CalculatorSwap the rows and columns of a matrix.
  • Matrix Rank CalculatorFind how many independent rows a matrix has.
  • Eigenvalue CalculatorFind the eigenvalues of a square matrix and see the characteristic polynomial.
  • 2x2 Matrix Determinant CalculatorEnter a, b, c and d to get ad − bc.
  • 3x3 Matrix Determinant CalculatorExpand a 3 × 3 determinant along the first row with every minor shown.
  • 3x3 Inverse Matrix CalculatorInvert a 3 × 3 matrix with every row operation listed.
  • RREF CalculatorRow-reduce any matrix up to 6 × 6 to reduced row echelon form.
  • Adjugate and Cofactor Matrix CalculatorGet the cofactor matrix, its transpose the adjugate, and the determinant.
  • LU Decomposition CalculatorFactor a square matrix into a lower and an upper triangular matrix, with exact fractions.

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