1. What is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental in mathematics and have applications in physics, engineering, computer science, and more.

2. Matrix Operations

This calculator supports the following operations:

• Addition: Element-wise addition of two matrices of the same dimensions
• Multiplication: Dot product of two matrices where columns of first matrix equals rows of second
• Transpose: Flips matrix over its diagonal (rows become columns and vice versa)
• Determinant: Scalar value computed from a square matrix (1x1, 2x2, or 3x3)

3. How to Use This Calculator

Steps:

1. Select matrix dimensions (up to 5x5)
2. Enter values for Matrix A
3. Select operation type
4. If operation requires Matrix B, enter its values
5. Click "Calculate" to see the result

4. Applications of Matrix Operations

Matrix operations are used in:

• Solving systems of linear equations
• Computer graphics and transformations
• Machine learning algorithms
• Quantum mechanics calculations
• Economic modeling

5. Frequently Asked Questions (FAQ)

Q1: Why can't I multiply these two matrices?
A: For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.

Q2: What is the identity matrix?
A: A square matrix with 1s on the diagonal and 0s elsewhere. Multiplying any matrix by the identity matrix leaves it unchanged.

Q3: Why is the determinant important?
A: The determinant can determine if a matrix is invertible (non-zero determinant) and is used in solving systems of equations.

Q4: What is matrix inversion?
A: The inverse of a matrix A is a matrix that when multiplied by A gives the identity matrix. Not all matrices have inverses.

Q5: Can I calculate eigenvalues with this calculator?
A: No, this calculator doesn't support eigenvalue calculation, which requires more advanced operations.