Matrix Determinant
Calculate the determinant of 2x2, 3x3, and larger square matrices. Get instant step-by-step solutions — completely free, no signup required.
Free Matrix Determinant Calculator
Enter your problem below to get an instant step-by-step solution.
About Matrix Determinant
Calculate the determinant of 2x2, 3x3, and larger square matrices. Linear algebra and matrix operations are foundational to computer graphics, machine learning, and solving systems of equations. Understanding matrix manipulations is critical for advanced mathematics.
Our Matrix Determinant calculator handles every type of problem in this topic. Whether you are a student working through homework, preparing for an exam, or a professional needing a quick calculation, you can enter any matrix determinant expression and receive a complete, verified solution with every step shown.
Matrix Determinant Formula
det [[a, b], [c, d]] = ad − bc
The determinant is a single number computed from a square matrix. It tells you whether the matrix is invertible (det ≠ 0), how it scales area or volume, and appears in Cramer’s rule. For 3×3 matrices, expand along a row using cofactors.
Worked Example
Find det [[2, −1, 0], [1, 3, 2], [0, 1, 4]]
- Expand along row 1: 2·det[[3, 2], [1, 4]] − (−1)·det[[1, 2], [0, 4]] + 0
- First minor: 3·4 − 2·1 = 10, so 2 × 10 = 20
- Second minor: 1·4 − 2·0 = 4, so +1 × 4 = 4
- Sum: 20 + 4 = 24
Answer: det = 24
Common Mistakes to Avoid
- Forgetting the alternating + − + sign pattern for cofactors.
- Computing ad + bc instead of ad − bc.
- Trying to take the determinant of a non-square matrix.
How to Use the Matrix Determinant Calculator
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Input your matrix or vector values for the Matrix Determinant problem
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Press Enter or click the Solve button to run the calculation
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Review the step-by-step solution that appears below the input
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Check the breakdown of each operation to understand the method
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Try a different problem to practice and build your understanding
Frequently Asked Questions
What is the Matrix Determinant formula?
det [[a, b], [c, d]] = ad − bc. The determinant is a single number computed from a square matrix. It tells you whether the matrix is invertible (det ≠ 0), how it scales area or volume, and appears in Cramer’s rule. For 3×3 matrices, expand along a row using cofactors.
Can you show a worked Matrix Determinant example?
Find det [[2, −1, 0], [1, 3, 2], [0, 1, 4]]: Expand along row 1: 2·det[[3, 2], [1, 4]] − (−1)·det[[1, 2], [0, 4]] + 0; First minor: 3·4 − 2·1 = 10, so 2 × 10 = 20; Second minor: 1·4 − 2·0 = 4, so +1 × 4 = 4; Sum: 20 + 4 = 24. Answer: det = 24.
Is this Matrix Determinant calculator free to use?
Yes — the Matrix Determinant calculator on MathSimplify is completely free. There is no signup, no subscription, and no usage limit. Enter any problem and get the step-by-step solution instantly.
Can I use this for Matrix Determinant homework and exam practice?
Absolutely. The step-by-step breakdown is designed specifically for learning. You can use it to check your work, understand where you went wrong, or study the method before an exam. The calculator shows the full working, not just the final answer.