3x3 Determinant Calculator

3x3 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. It provides important information about the matrix, such as whether it is invertible and the scaling factor of the linear transformation it represents.

2. How Does the Calculator Work?

The calculator uses the standard 3x3 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 for solving systems of linear equations, finding inverses of matrices, calculating eigenvalues, and determining whether a matrix is singular or invertible.

4. Using the Calculator

Tips: Enter all nine values of the 3x3 matrix (a through i) in the corresponding input fields. The calculator will compute the determinant using the standard formula.

5. Frequently Asked Questions (FAQ)

Q1: What does a zero determinant indicate?
A: A determinant of zero indicates that the matrix is singular (not invertible) and the system of equations it represents has either no solution or infinitely many solutions.

Q2: Can the determinant be negative?
A: Yes, determinants can be negative. The sign indicates the orientation of the transformation (positive preserves orientation, negative reverses it).

Q3: What is the geometric interpretation of the determinant?
A: For a 2x2 matrix, the absolute value of the determinant represents the area scaling factor. For a 3x3 matrix, it represents the volume scaling factor of the linear transformation.

Q4: Are there other methods to calculate 3x3 determinants?
A: Yes, besides the standard formula, you can use the rule of Sarrus, cofactor expansion along any row or column, or triangularization methods.

Q5: When is the determinant calculation used in real applications?
A: Determinants are used in engineering, physics, computer graphics, economics, and many other fields for solving systems of equations, calculating areas/volumes, and analyzing linear transformations.