A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. It is used to represent mathematical objects or properties of such objects.

How do I multiply two matrices?

To multiply two matrices, you perform the dot product of the rows of the first matrix with the columns of the second matrix. The number of columns in the first matrix must equal the number of rows in the second matrix.

Is matrix multiplication commutative?

In general, matrix multiplication is not commutative. This means that A × B is usually not equal to B × A, unlike regular number multiplication.

Can I add matrices of different sizes?

No, you can only add or subtract matrices that have the exact same dimensions (same number of rows and columns).

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Matrix Calculator

Instantly add, subtract, and multiply 2x2 matrices. A free tool for students and professionals to verify matrix operations.

Matrix A (2×2)

Matrix B (2×2)

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Reviewed by Dr. Alex M., Ph.D.

Mathematics Professor & Data Scientist | Last Updated: November 2025

Understanding Matrix Operations

Matrices are fundamental tools in mathematics, physics, and computer science. They are used to solve systems of linear equations, represent geometric transformations, and much more. This calculator helps you perform basic operations on 2x2 matrices.

How to Add and Subtract Matrices

Adding or subtracting matrices is straightforward. You simply add or subtract the corresponding elements of the two matrices. For this to work, both matrices must have the same dimensions.

For example, if you have Matrix A and Matrix B:

(A + B)_ij = A_ij + B_ij
(A - B)_ij = A_ij - B_ij

How to Multiply Matrices

Matrix multiplication is a bit more complex. It involves calculating the dot product of the rows of the first matrix with the columns of the second matrix.

For a 2x2 matrix multiplication (C = A × B):

C₁₁ = A₁₁B₁₁ + A₁₂B₂₁
C₁₂ = A₁₁B₁₂ + A₁₂B₂₂
C₂₁ = A₂₁B₁₁ + A₂₂B₂₁
C₂₂ = A₂₁B₁₂ + A₂₂B₂₂

💡 Expert Tip: Order Matters!

Unlike regular multiplication where 3 × 5 is the same as 5 × 3, in matrix multiplication, A × B is usually not the same as B × A. This property is known as non-commutativity. Always pay attention to the order of multiplication.

⚠️ Common Mistake: Dimension Mismatch

A common error is trying to multiply matrices with incompatible dimensions. Remember: The number of columns in the first matrix must equal the number of rows in the second matrix. For 2x2 matrices, this is always satisfied.

Frequently Asked Questions

What is the identity matrix?

The identity matrix is a special square matrix with ones on the main diagonal and zeros elsewhere. When you multiply any matrix by the identity matrix, the result is the original matrix unchanged.

Can I divide matrices?

Technically, there is no such thing as matrix division. Instead, we multiply by the inverse of a matrix. This is similar to multiplying by the reciprocal of a number.

What are matrices used for in real life?

Matrices are used in computer graphics to rotate and scale images, in physics to describe quantum mechanics, in economics to model markets, and in cryptography to encrypt data.

References

1. "Matrix (mathematics)." Wikipedia. https://en.wikipedia.org/wiki/Matrix_(mathematics)
2. "Matrix Multiplication." Khan Academy. https://www.khanacademy.org/math/precalculus