Matrix Calculator — Step-by-Step Solutions

Your calculations will appear here

The Matrix Calculator performs common matrix operations including addition, subtraction, multiplication, determinant calculation, transpose, and row reduction to RREF. Supports 2x2, 3x3, and 4x4 matrices.

Enter your matrix values and select an operation to get instant results with step-by-step working shown in mathematical notation.

How to Use the Matrix Calculator

Select the matrix operation (Add, Multiply, Determinant, Transpose, or RREF).
Choose the matrix size (2x2, 3x3, or 4x4).
Enter the values for each cell in the matrix grid.
Click Calculate to see the result with step-by-step working.

Formula and Theory

Matrices are rectangular arrays of numbers arranged in rows and columns. They are fundamental to linear algebra and used extensively in physics, engineering, computer graphics, and data science.

Matrix addition and subtraction require matrices of the same dimensions and operate element-wise. Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second.

The determinant is a scalar value computed from a square matrix that indicates whether the matrix is invertible. A zero determinant means the matrix is singular (not invertible).

Row Echelon Form (RREF) is achieved through Gaussian elimination and is used to solve systems of linear equations, find matrix rank, and determine linear independence.

Worked Examples

Example 1: Matrix Addition

Problem: Add [[1,2],[3,4]] + [[5,6],[7,8]]

Solution: [[1+5, 2+6], [3+7, 4+8]]

Answer: [[6, 8], [10, 12]]

Example 2: 2x2 Determinant

Problem: Find det([[3,1],[2,5]])

Solution: det = (3)(5) - (1)(2) = 15 - 2

Answer: 13

Example 3: Matrix Transpose

Problem: Transpose [[1,2,3],[4,5,6]]

Solution: Swap rows and columns

Answer: [[1,4],[2,5],[3,6]]

Example 4: Matrix Multiplication

Problem: [[1,2],[3,4]] x [[5,6],[7,8]]

Solution: [[1*5+2*7, 1*6+2*8], [3*5+4*7, 3*6+4*8]]

Answer: [[19, 22], [43, 50]]

Frequently Asked Questions

What is the determinant of a matrix?

The determinant is a scalar value that can be computed from a square matrix. It tells you whether the matrix is invertible (non-zero determinant) and is used in solving systems of equations, finding eigenvalues, and computing volumes.

When can you multiply two matrices?

You can multiply matrix A (m x n) by matrix B (p x q) only when n = p. The result is an m x q matrix.

What is RREF used for?

Row Reduced Echelon Form is used to solve systems of linear equations, find the rank of a matrix, determine linear independence, and find the inverse of a matrix.

Is matrix multiplication commutative?

No. In general, AB is not equal to BA. This is one of the key differences between matrix and scalar arithmetic.

Related Linear Algebra Calculators

eigenvalue

Eigenvalue Calculator

Calculate eigenvalues and eigenvectors of matrices. Supports 2x2 and 3x3 matrices. Free online eigenvalue calculator.

matrix

Cross Product Calculator

Calculate the cross product of two 3D vectors. Find magnitude, unit vector direction, parallel check, and dot product. Free online vector calculator.

matrix

Determinant Calculator

Calculate the determinant of 2x2, 3x3, and 4x4 matrices. Step-by-step cofactor expansion with detailed workings. Free online matrix determinant calculator.

matrix

Null Space Calculator

Find the null space basis, nullity, RREF with free variables, and column space of a matrix. Free online null space calculator with step-by-step Gaussian elimination.

matrix