Free Matrix Calculator
Enter two matrices and get the product, inverse, determinant or transpose with every intermediate row shown. Built for Class 12 CBSE/ICSE, A-Level Further Maths and JEE Main linear algebra.
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A worked 2×2 matrix multiplication — element by element
Below is one fully worked example plus a short primer so you can see exactly how our AI reasons through a problem.
Example Problem
DEMO- 1
Check the dimensions are compatible
A is 2×2, B is 2×2. For the product AB to exist, the number of columns of A must equal the number of rows of B — here both are 2, so AB exists and is 2×2.
- 2
Row 1, column 1 of AB
Dot product of A's row 1 with B's column 1.
- 3
Row 1, column 2 of AB
A's row 1 · B's column 2.
- 4
Row 2, column 1 of AB
A's row 2 · B's column 1.
- 5
Row 2, column 2 of AB
A's row 2 · B's column 2.
Final Answer
How matrix multiplication and inversion work — rules and shortcuts
A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions are written as rows × columns — a matrix with 2 rows and 3 columns is 2×3, pronounced 'two by three'. Matrices are the foundation of linear algebra and are essential for Class 12 CBSE/ICSE Maths, JEE Main and Advanced, A-Level Further Maths, and every engineering or data-science course that follows.
School syllabuses focus on five operations. Addition and subtraction — simple element-wise operation; both matrices must have identical dimensions. Scalar multiplication — multiply every element by the scalar. Matrix multiplication — the row-column dot product shown in the example; dimensions must match (columns of the first = rows of the second), and crucially AB ≠ BA in general (non-commutative). Transpose — flip the matrix across its main diagonal, written Aᵀ. Determinant — a single scalar computed from a square matrix; non-zero means the matrix is invertible, zero means it's singular.
The inverse A⁻¹ exists only when det(A) ≠ 0, and satisfies A · A⁻¹ = I (the identity matrix). For a 2×2 matrix [[a, b], [c, d]] the inverse is (1/det) × [[d, −b], [−c, a]] where det = ad − bc. For 3×3 matrices, use the adjugate method: A⁻¹ = adj(A) / det(A), where adj(A) is the transpose of the cofactor matrix. Both methods scale to any square size but the adjugate method gets tedious past 4×4 — row reduction (Gauss-Jordan) is more practical beyond that.
The solver above handles matrices up to 4×4 quickly, with every intermediate step — row reductions for inverses, cofactor expansions for determinants, and full dot products for multiplication. You can paste matrices in standard formats ([[1,2],[3,4]]) or type them with LaTeX. Eigenvalues and eigenvectors are also supported for 2×2 and 3×3 matrices, which is the JEE Advanced scope.
Matrix problems to practise (Class 12, A-Level)
Tap any problem to solve it with full step-by-step working.
- Solve with AI →1.DeterminantClass 12Easy
- Solve with AI →2.Inverse (2×2)Class 12Medium
- Solve with AI →3.Matrix equationClass 12Medium
- Solve with AI →4.Determinant (3×3)Class 12 / JEEHard
- Solve with AI →5.EigenvaluesJEE AdvancedHard
Frequently asked questions
How do I multiply two matrices step by step?+
First check dimensions: columns of the first matrix must equal rows of the second. For each element of the result, take the dot product of the corresponding row of the first matrix with the corresponding column of the second — multiply element-wise and sum. The result has the same number of rows as the first matrix and the same number of columns as the second.
When does a matrix have an inverse?+
A square matrix has an inverse if and only if its determinant is non-zero. If det(A) = 0, the matrix is called singular and has no inverse. Non-square matrices never have an inverse in the usual sense (though they may have pseudo-inverses — beyond school scope).
How do I find the determinant of a 3×3 matrix?+
Use cofactor expansion along any row or column: multiply each element by its signed cofactor (the 2×2 determinant obtained by deleting its row and column, with a ± sign by checkerboard pattern), then sum. Usually expand along the row or column with the most zeros to minimise work. The solver shows the expansion explicitly.
Why is matrix multiplication not commutative?+
AB and BA usually give different results — try A = [[1,2],[0,1]] and B = [[1,0],[1,1]] and check. This is because the row-column dot product treats the two matrices asymmetrically: rows of the first, columns of the second. The only matrices that always commute are scalar multiples of the identity.
What's the difference between the transpose and the inverse?+
Transpose (Aᵀ) flips the matrix across its main diagonal — element (i,j) becomes element (j,i). It always exists and is easy to compute. The inverse (A⁻¹) is the unique matrix such that A·A⁻¹ = I, only exists when det(A) ≠ 0, and requires cofactors or row reduction to compute.
Can I use matrices to solve simultaneous equations?+
Yes. Write the system Ax = b, where A is the coefficient matrix, x is the vector of unknowns, b is the constant vector. If A is invertible, x = A⁻¹b. This is called the matrix method and is the standard Class 12 and A-Level approach for 3+ variable systems.
Does the solver handle 4×4 or 5×5 matrices?+
4×4 is supported with full step-by-step working — determinants by cofactor expansion, inverses by adjugate method or row reduction, multiplication element by element. Beyond 4×4 the output stays correct but the step-by-step becomes very long; most school and entrance-exam problems top out at 3×3 anyway.
What are eigenvalues and are they covered?+
An eigenvalue λ of a matrix A satisfies A·v = λ·v for some non-zero vector v (the eigenvector). Covered for 2×2 and 3×3 matrices — JEE Advanced and A-Level Further Maths scope. The solver finds eigenvalues by computing the characteristic polynomial det(A − λI) = 0 and solving for λ, then back-solves for each eigenvector.
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