SILDIL

Matrix Calculator

Calculate matrix addition, subtraction, multiplication, determinants, inverses, transpose, and more with customizable rows and columns and step-by-step solutions.

16 0

Matrix A

Dimensions: 2 × 2

Matrix B

Dimensions: 2 × 2

Quick examples

Result
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Dimensions: —

How to Use the Matrix Calculator

Step-by-step method to perform matrix operations, find determinants, inverses, and more with customizable dimensions.

  1. 1. Choose an operation: Select addition, subtraction, multiplication, inverse, determinant, transpose, trace, or rank from the operation menu.
  2. 2. Set matrix dimensions: Set the number of rows and columns for Matrix A (and Matrix B when the operation requires it).
  3. 3. Enter your values: Type numbers into each cell. You can use arrow keys to quickly move between adjacent cells.
  4. 4. Review results and steps: View the live result, resulting dimensions, and a step-by-step breakdown of how the calculation was performed.

What Is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used throughout linear algebra, computer graphics, statistics, physics, and engineering.

How to Use the Matrix Calculator

  1. Choose an operation (addition, multiplication, inverse, and more).
  2. Set the number of rows and columns for Matrix A (and Matrix B when needed).
  3. Enter values in each cell. Arrow keys move between adjacent cells.
  4. View the live result, dimensions, and calculation steps.

Matrix Multiplication

If A is m × n and B is n × p, then AB is m × p. The columns of A must match the rows of B. Each entry of the product is a dot product of a row of A with a column of B.

Matrix multiplication is not commutative in general: AB ≠ BA.

Matrix Addition and Subtraction

Addition and subtraction require identical dimensions. Each result entry is the sum or difference of the corresponding entries.

Matrix Inverse

Only square matrices can have an inverse. A is invertible when det(A) ≠ 0. Then A × A⁻¹ = A⁻¹ × A = I, the identity matrix. Singular matrices (determinant zero) have no inverse.

How to Find the Determinant

For a 2×2 matrix [[a,b],[c,d]], det = ad − bc. Larger matrices use Gaussian elimination with partial pivoting for numerical stability.

Matrix Transpose

The transpose Aᵀ swaps rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ. An m×n matrix becomes n×m.

Matrix Rank

Rank is the number of linearly independent rows (or columns), equal to the number of nonzero pivot rows after row reduction. Rank applies to rectangular matrices as well as square ones.

3 × 3 Matrix Inverse

Enter any invertible 3×3 matrix, choose Matrix Inverse, and review the determinant, inverse, and verification that A × A⁻¹ ≈ I.

4 × 4 Matrix Calculator

All operations support up to 10×10 matrices, including full 4×4 addition, multiplication, determinant, inverse, transpose, trace, and rank.

Common Matrix Calculator Mistakes

  • Multiplying matrices with incompatible dimensions (columns of A ≠ rows of B).
  • Adding matrices of different sizes.
  • Trying to invert a singular matrix.
  • Confusing transpose with inverse.
  • Assuming matrix multiplication is commutative.

Frequently Asked Questions

Q: What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used throughout linear algebra, computer graphics, statistics, physics, and engineering.

Q: How do you multiply two matrices?

If A is m × n and B is n × p, then AB is m × p. The columns of A must match the rows of B. Each entry of the product is a dot product of a row of A with a column of B.

Q: Can you add or subtract matrices of different sizes?

No. Addition and subtraction require identical dimensions. Each result entry is simply the sum or difference of the corresponding entries.

Q: What is a matrix inverse and when does it exist?

Only square matrices can have an inverse. A matrix A is invertible when det(A) ≠ 0. Then A × A⁻¹ = A⁻¹ × A = I (the identity matrix). Singular matrices (determinant zero) have no inverse.

Q: How do you find the determinant of a matrix?

For a 2×2 matrix [[a,b],[c,d]], det = ad − bc. Larger matrices use Gaussian elimination with partial pivoting for numerical stability.

Q: What is the transpose of a matrix?

The transpose Aᵀ swaps rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ. An m×n matrix becomes an n×m matrix.

Q: What is the trace of a matrix?

The trace is the sum of the elements on the main diagonal (from the upper left to the lower right) of a square matrix.

Q: What is the rank of a matrix?

Rank is the number of linearly independent rows (or columns), equal to the number of nonzero pivot rows after row reduction. It applies to rectangular matrices as well as square ones.

Q: Why is matrix multiplication not commutative?

In general, AB ≠ BA. The order of multiplication matters because the dot products of rows and columns yield different results depending on which matrix provides the rows and which provides the columns.

Q: Can this calculator handle 3x3 or 4x4 matrices?

Yes. All operations support up to 10×10 matrices, including full 3×3 and 4×4 addition, multiplication, determinant, inverse, transpose, trace, and rank.

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