Free Matrix Calculator with Steps
Calculate determinants, inverses, RREF, and more step-by-step. Learn linear algebra with instant feedback. Perfect for A-Level and university maths.
Step-by-step learning with explanations
Select Operation
Enter Matrices
Fill in the matrix values
Matrix A
Matrix B
Select an operation and enter matrix values, then click Calculate
Understanding Matrix Operations
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental in linear algebra and have applications in graphics, physics, engineering, and computer science.
The determinant of a square matrix is a scalar value that encodes important properties - a non-zero determinant means the matrix is invertible. The rank tells us the number of linearly independent rows or columns.
Basic Operations
Key Matrix Concepts
Matrix Multiplication
Result C = A × B has size m×p
C[i][j] = Σ(A[i][k] × B[k][j])
Columns of A = Rows of B
Determinant & Inverse
det(A) ≠ 0 ⟹ A is invertible
A × A⁻¹ = I (identity matrix)
A⁻¹ = (1/det) × adj(A)
RREF & Rank
Use Gaussian elimination
Leading 1s in each pivot position
Rank = number of non-zero rows
How It Works
Select Operation
Choose from add, subtract, multiply, determinant, inverse, RREF, or rank.
Set Dimensions
Choose matrix size (rows × columns) up to 5×5.
Enter Values
Input numbers into the matrix cells.
Learn or Get Answer
Learn Mode teaches each step; Quick Mode gives instant results.
Frequently Asked Questions
How do I calculate a determinant?
For 2×2 matrices: det = ad - bc. For larger matrices, use cofactor expansion along any row or column, multiplying elements by their cofactors.
How do I find the inverse of a matrix?
Use A⁻¹ = (1/det(A)) × adj(A). The matrix must be square with non-zero determinant. The adjugate is the transpose of the cofactor matrix.
How do I multiply two matrices?
Multiply each element in row i of A by the corresponding element in column j of B, then sum. The number of columns in A must equal rows in B.
What is RREF used for?
RREF (Row Reduced Echelon Form) is used to solve systems of linear equations, find matrix rank, and determine if a matrix is invertible.
What does matrix rank tell us?
Rank is the number of linearly independent rows/columns. It tells you the dimension of the solution space and whether a system has unique, infinite, or no solutions.
Is this suitable for A-Level maths?
Yes! Covers all A-Level and Further Maths matrix topics including determinants, inverses, multiplication, and solving simultaneous equations using matrices.
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