Matrix Subtraction Calculator
Subtract two matrices of the same dimensions. Enter your matrices below and get instant results with step-by-step solutions.
Matrix A
Matrix B
Result
How to Subtract Matrices
Same Dimensions
Matrices must have the same number of rows and columns to be subtracted.
Element-wise Subtraction
Subtract corresponding elements from each matrix to get the result matrix.
Non-Commutative
Matrix subtraction is NOT commutative: A - B ≠ B - A
Relation to Addition
A - B = A + (-B), where -B is the additive inverse of B
Mathematical Theory & History
What is Matrix Subtraction?
Matrix subtraction is a fundamental operation in linear algebra where corresponding elements of the second matrix are subtracted from the first matrix. Both matrices must have identical dimensions.
Formula: (A - B)ij = Aij - Bij
Where A and B are matrices of the same size, and i,j represent the row and column indices.
Historical Background
Matrix subtraction, like addition, was formalized as part of matrix algebra by Arthur Cayley in his groundbreaking 1858 work. The operation follows naturally from the definition of matrix addition and the concept of additive inverses.
Cayley's systematic approach to matrix operations established subtraction as A - B = A + (-B), where -B represents the additive inverse of matrix B. This framework became essential for solving systems of linear equations and modern computational mathematics.
Properties of Matrix Subtraction
Non-Commutative
A - B ≠ B - A (in general)
Non-Associative
(A - B) - C ≠ A - (B - C)
Identity Element
A - 0 = A (zero matrix)
Self-Subtraction
A - A = 0 (zero matrix)
Real-World Applications
Image Processing
Background subtraction for motion detection and object tracking
Economics
Calculating differences between economic models and scenarios
Engineering
Stress analysis by subtracting baseline measurements from loaded states
Data Analysis
Computing differences between datasets for trend analysis
Frequently Asked Questions
No, matrix subtraction is NOT commutative. A - B is generally not equal to B - A. The order matters in subtraction, unlike addition where A + B = B + A always holds.
No, just like addition, matrix subtraction requires both matrices to have exactly the same dimensions. You cannot subtract a 2×3 matrix from a 3×2 matrix, for example.
When you subtract a matrix from itself (A - A), the result is always the zero matrix of the same dimensions, where every element equals zero. This is analogous to any number minus itself equaling zero.
Matrix subtraction can be expressed as addition: A - B = A + (-B), where -B is the additive inverse of B (every element of B multiplied by -1). This relationship helps in understanding and computing subtraction.
Yes, you can perform sequential subtractions like A - B - C, but remember that subtraction is not associative. (A - B) - C is different from A - (B - C). Always work from left to right unless parentheses indicate otherwise.
Matrix subtraction IS element-wise subtraction. Each element in the result is obtained by subtracting the corresponding element of the second matrix from the first matrix. This is different from other matrix operations like multiplication.
Negative numbers follow standard arithmetic rules. Subtracting a negative number is equivalent to adding its positive value: a - (-b) = a + b. Our calculator handles all combinations of positive and negative numbers automatically.