Matrix Calculator

Matrix Operations Guide

A matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are fundamental tools in linear algebra, physics, computer graphics, and engineering.

Fundamental Matrix Operations

1. Matrix Addition and Subtraction

You can add or subtract matrices only if they have the same dimensions (same number of rows and columns). You perform the operation element by element.

Example:

$$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$$

2. Matrix Multiplication

To multiply two matrices $A$ (size $m \times n$) and $B$ (size $n \times p$), the number of columns in $A$ must equal the number of rows in $B$. The resulting matrix will have size $m \times p$.

The element at row $i$ and column $j$ of the product is the dot product of row $i$ from $A$ and column $j$ from $B$.

3. Determinant

The determinant is a scalar value calculated from a square matrix (e.g., $2 \times 2$ or $3 \times 3$). It provides important properties about the matrix, such as whether it is invertible.

For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$: $\det(A) = ad - bc$

If $\det(A) = 0$, the matrix is singular and has no inverse.

4. Matrix Inverse

If $A$ is a square matrix and has a non-zero determinant, its inverse $A^{-1}$ exists such that: $A \times A^{-1} = I$ where $I$ is the identity matrix.

5. Transpose

The transpose of a matrix $A$ (denoted $A^T$ or $A'$) is formed by flipping rows and columns. The element at position $(i, j)$ becomes the element at $(j, i)$.

Real-World Applications

• Computer Graphics: Matrices are used to rotate, scale, and translate 3D objects on a screen.
• Cryptography: Matrices are used to encrypt and decrypt messages.
• Network Theory: Adjacency matrices represent connections in networks (e.g., social graphs).
• Physics: Used in quantum mechanics and general relativity.

Frequently Asked Questions

Q: Can you divide matrices? A: Technically, no. There is no concept of matrix division. Instead, you multiply by the inverse of a matrix. To "divide" $A$ by $B$, you compute $A \times B^{-1}$.

Q: Is matrix multiplication commutative? A: Generally, no. $A \times B$ is typically NOT equal to $B \times A$. The order of multiplication matters significantly.

Q: What is an identity matrix? A: An identity matrix ($I$) is a square matrix with 1s on the main diagonal and 0s everywhere else. It acts like the number 1 in regular multiplication: $A \times I = A$.