Matrix Calculator

Q: What is a matrix calculator used for?

A matrix calculator is a tool used to perform complex operations on matrices, which are rectangular arrays of numbers. It is widely used in linear algebra, computer graphics, physics, engineering, and data science to solve systems of linear equations, perform geometric transformations, and analyze data.

Q: What makes a matrix singular?

A square matrix is called 'singular' if its determinant is equal to zero. A singular matrix does not have an inverse. This calculator will return an error if you try to find the inverse of a singular matrix.

Q: Is matrix multiplication commutative?

No, matrix multiplication is not commutative. This means that in most cases, A × B is not equal to B × A. The order of multiplication matters.

Q: What is an identity matrix?

An identity matrix (denoted as 'I') is a square matrix with ones on the main diagonal and zeros everywhere else. It is the matrix equivalent of the number 1. Any matrix multiplied by an identity matrix (of the correct size) results in the original matrix.

This comprehensive matrix calculator performs a wide range of operations on matrices, including addition, subtraction, multiplication, finding the determinant, and calculating the inverse. Set the dimensions of Matrix A and Matrix B, fill in the values, and select an operation to get started.

Matrix A

Rows: Cols:

Matrix B

Rows: Cols:

Operations

Result

How to Use the Matrix Calculator

Set Dimensions: Adjust the "Rows" and "Cols" for Matrix A and Matrix B to your desired size. The input grids will update automatically (max 10x10).
Fill in Values: Enter the elements for each matrix. You can use integers (5), decimals (2.5), or fractions (1/2).
Select Operation: Click the button for the operation you want to perform (e.g., [A + B], [A × B], [det(A)]).
View Result: The result will be displayed in the "Result" section. If the operation is mathematically impossible (like adding matrices of different sizes), an error message will appear.

Understanding Matrix Operations

A matrix is a rectangular array of numbers arranged in rows and columns. This calculator supports the fundamental operations of linear algebra.

Matrix Addition and Subtraction (A ± B)

To add or subtract two matrices, they must have the **exact same dimensions**. The operation is performed by adding or subtracting the corresponding elements.

$(A + B)_{ij} = A_{ij} + B_{ij}$
$(A - B)_{ij} = A_{ij} - B_{ij}$

Matrix Multiplication (A × B)

To multiply Matrix A (dimensions $m \times n$) by Matrix B (dimensions $p \times q$), the number of **columns in A ($n$) must equal the number of rows in B ($p$)**. The resulting matrix will have dimensions $m \times q$.

$(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$

This means each element $(AB)_{ij}$ is the dot product of the $i$-th row of A and the $j$-th column of B. Note that matrix multiplication is **not commutative** (A × B ≠ B × A).

Determinant of a Matrix (det(A))

The determinant is a scalar value that can only be computed for a **square matrix** (e.g., 2x2, 3x3). It provides important information about the matrix, such as whether it is invertible.

For a 2x2 matrix: $\text{det}(A) = ad - bc$
For a 3x3 matrix (Cofactor Expansion):
$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$

Inverse of a Matrix (A⁻¹)

The inverse of a matrix A, denoted $A⁻¹$, is a matrix such that $A \times A⁻¹ = I$ (the identity matrix). An inverse only exists if the matrix is **square** and **non-singular** (its determinant is not zero).

$A^{-1} = \frac{1}{\text{det}(A)} C^T$

Where $C^T$ is the transpose of the cofactor matrix (also known as the adjugate or classical adjoint of A). If $\text{det}(A) = 0$, the matrix is singular and has no inverse.

Transpose of a Matrix (Aᵀ)

The transpose of a matrix is found by swapping its rows and columns. The element at $A_{ij}$ becomes the element at $(A^T)_{ji}$.