Matrix Calculator

A matrix calculator is an online tool that performs various mathematical operations on matrices, including addition, subtraction, multiplication, finding determinants, inverses, and transposes. It helps students, engineers, and researchers solve linear algebra problems quickly and accurately.

To multiply two matrices A and B, the number of columns in matrix A must equal the number of rows in matrix B. The resulting matrix will have the same number of rows as A and the same number of columns as B. Each element is calculated using the dot product of the corresponding row and column.

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it's invertible (determinant ≠ 0) or singular (determinant = 0). For a 2×2 matrix, det(A) = ad - bc.

A matrix inverse A⁻¹ exists only for square matrices with a non-zero determinant. The inverse satisfies the equation A × A⁻¹ = I, where I is the identity matrix. For larger matrices, methods like Gaussian elimination or adjugate matrix division are used.

Matrix transpose is an operation that flips a matrix over its diagonal, switching row and column indices. If A is an m×n matrix, then Aᵀ (A transpose) is an n×m matrix where the element at position (i,j) in A becomes the element at position (j,i) in Aᵀ.

No, matrix addition and subtraction can only be performed on matrices of the same dimensions. Both matrices must have exactly the same number of rows and columns. If the dimensions don't match, the operation is undefined.

The rank of a matrix is the maximum number of linearly independent rows (or columns) in the matrix. It represents the dimension of the vector space spanned by the matrix's rows or columns. The rank is always less than or equal to the minimum of the number of rows and columns.