Inverse Matrix Calculator

In linear algebra, finding the inverse of a matrix is a fundamental operation used in solving systems of equations, computer graphics, physics, and engineering. The inverse matrix $A^{-1}$A−1 is defined such that:$$A \cdot A^{-1} = I$$A⋅A−1=I

where $I$I is the identity matrix.

Our Inverse Matrix Calculator allows you to quickly compute the inverse of 2x2 and 3x3 matrices, as well as the determinant, without manual calculations. This tool is perfect for students, teachers, and professionals alike.

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How to Use the Inverse Matrix Calculator

1. Select the Matrix Size:
   Choose 2x2 or 3x3 from the dropdown menu.
2. Enter Matrix Values:
   Fill in each cell of the matrix with numbers. Leave a cell blank for 0.
3. Calculate Inverse:
   Click the Calculate button. The calculator will display:
   • Determinant – shows if the matrix is invertible
   • Inverse Matrix – calculated with high precision
4. Reset for New Matrix:
   Click Reset to clear inputs and start over.

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How It Works

• 2x2 Matrix Formula:
  For a matrix

$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$A=[ac​bd​]

The inverse is:$$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$A−1=ad−bc1​[d−c​−ba​]

• 3x3 Matrix Formula:
  For a 3x3 matrix, the inverse is computed using:

1. Determinant
2. Cofactor matrix
3. Adjugate (transpose of cofactor)
4. Divide each element by the determinant

If the determinant is 0, the matrix is singular and has no inverse.

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Example Calculations

Example 1: 2x2 Matrix

Matrix:$$\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$$[21​34​]

Result:

• Determinant = 5
• Inverse Matrix =

$$\begin{bmatrix} 0.8000 & -0.6000 \\ -0.2000 & 0.4000 \end{bmatrix}$$[0.8000−0.2000​−0.60000.4000​]

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Example 2: 3x3 Matrix

Matrix:$$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$$​105​216​340​​

Result:

• Determinant = 1
• Inverse Matrix =

$$\begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{bmatrix}$$​−2420−5​18−154​5−41​​

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Benefits of Using This Calculator

• Quick and Accurate: Avoid lengthy hand calculations.
• Educational: Helps visualize cofactor, adjugate, and determinant steps.
• User-Friendly: Simple interface for students and professionals.
• Versatile: Handles both 2x2 and 3x3 matrices efficiently.
• Error-Proof: Alerts if a matrix is singular and cannot be inverted.

This tool is ideal for homework, linear algebra exercises, engineering computations, and computer graphics problems.

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FAQs

1. What is an inverse matrix?
   The inverse $A^{-1}$A−1 is a matrix that, when multiplied with the original matrix $A$A, gives the identity matrix.
2. Can I invert a singular matrix?
   No, a matrix with determinant 0 cannot be inverted.
3. Does it support decimals?
   Yes, all numbers, including decimals, are supported.
4. Can I calculate both 2x2 and 3x3 matrices?
   Yes, the calculator supports both sizes.
5. What is the determinant?
   The determinant indicates whether the matrix is invertible. A nonzero determinant means the inverse exists.
6. Does it round results?
   Yes, results are rounded to 4 decimal places for clarity.
7. Is it suitable for students?
   Absolutely. It’s perfect for homework and learning matrix operations.
8. Can I reset the matrix?
   Yes, the Reset button clears all inputs.
9. Is it free?
   Yes, the calculator is completely free to use.
10. How accurate is it?
    It uses precise arithmetic and rounding only for display purposes.

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Conclusion

The Inverse Matrix Calculator is a reliable tool for quickly determining the inverse and determinant of 2x2 and 3x3 matrices. It saves time, reduces errors, and helps users understand the underlying linear algebra concepts.

Whether for education, research, or professional applications, this calculator provides accurate results in seconds.