3 Matrix Determinant Calculator

3x3 Matrix Determinant Formula:

\[ \det = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Value a:

Value b:

Value c:

Value d:

Value e:

Value f:

Value g:

Value h:

Value i:

Determinant:

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1. What is a 3x3 Matrix Determinant?

The determinant of a 3x3 matrix is a scalar value that can be computed from the elements of the matrix and provides important information about the matrix properties, such as whether it is invertible or singular.

2. How Does the Calculator Work?

The calculator uses the standard 3x3 matrix determinant formula:

\[ \det = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Where the matrix is arranged as: \[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]

Explanation: The formula calculates the determinant by expanding along the first row using the rule of Sarrus or cofactor expansion method.

3. Importance of Determinant Calculation

Details: Determinants are fundamental in linear algebra and have applications in solving systems of linear equations, finding inverses of matrices, calculating eigenvalues, and in various geometric transformations.

4. Using the Calculator

Tips: Enter all nine values of the 3x3 matrix in the corresponding input fields. The calculator will compute and display the determinant value.

5. Frequently Asked Questions (FAQ)

Q1: What does a zero determinant indicate?
A: A determinant of zero indicates that the matrix is singular, meaning it does not have an inverse and the system of equations it represents may have no solution or infinitely many solutions.

Q2: Can the determinant be negative?
A: Yes, determinants can be negative. The sign of the determinant indicates the orientation of the transformation represented by the matrix.

Q3: What is the geometric interpretation of determinant?
A: For a 3x3 matrix, the absolute value of the determinant represents the scaling factor of the volume when the matrix is applied as a linear transformation.

Q4: Are there other methods to calculate 3x3 determinants?
A: Yes, besides the standard formula, you can use the rule of Sarrus or reduce the matrix to upper triangular form and multiply the diagonal elements.

Q5: What are the applications of determinants?
A: Determinants are used in solving systems of equations, finding matrix inverses, calculating areas/volumes, in computer graphics, physics, and engineering applications.