Matrix Operations Calculator

Operation: Matrix A size: × Matrix B size: ×

Hint: For inverse we use Gauss–Jordan elimination; for 2×2 matrices the closed form A⁻¹ = (1/det)·[[d, -b],[-c, a]] is shown when applicable.

A Matrix Operations Calculator streamlines intricate matrix calculations by quickly performing addition, multiplication, determinants, and inverses. Perfect for both students and professionals, it provides rapid and precise outcomes, enhancing efficiency in linear algebra, engineering, data science, and scientific computing activities.

What Is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns.

Example:

A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]

Each element in a matrix can represent quantities such as coefficients, data points, or transformations.

What Are Matrix Operations?

Matrix operations refer to mathematical processes performed on matrices, such as:

Matrix Addition
Matrix Multiplication
Matrix Inverse

These operations help combine, transform, or analyze data represented in matrix form.

Matrix Operation Formulas

Matrix Addition

Two matrices can be added only if they have the same dimensions (same number of rows and columns).

Formula:

For matrices $A = [a_{i j}]$ and $B = [b_{i j}]$ :

A + B = C = [c_{i j}] = [a_{i j} + b_{i j}]

Matrix Multiplication

Matrix multiplication is defined when the number of columns in the matrix A equals the number of rows in the matrix B.

Formula:

If:

A = (m \times n), B = (n \times p)

Then the product $C = A \cdot B$ is:

c_{i j} = \sum_{k = 1} \overset{n}{} a_{i k} \cdot b_{k j}

Inverse of a Matrix

Only square matrices (same number of rows and columns) may have an inverse.

A matrix $A$ has an inverse $A^{- 1}$ if:

A \cdot A^{- 1} = I

where $I$ is the identity matrix.

Formula (2×2 Matrix)

A = [\begin{matrix} a & b \\ c & d \end{matrix}]

A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]

How to Use Our Matrix Operations Calculator

Select the desired “Operation” to be performed.
Select the size of “Matrix A” and “Matrix B”.
Click “Calculate,” and the calculator automatically computes the results.