Matrix Subtraction 4×4

Online calculator for subtracting two 4x4 matrices

Matrix Subtraction Calculator

Instructions

Enter the values of both matrices to be subtracted. Empty fields are counted as zero. Click Calculate.

Matrix A
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Matrix B
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Result: A − B
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Matrix Subtraction - Overview

Prerequisites

To subtract matrices, the matrices must match. That is, they must have the same number of rows and columns. Both matrices must be 4×4.

Calculation Formula

In order to perform a matrix subtraction, the corresponding matrix elements are subtracted:

\(C_{ij} = A_{ij} - B_{ij}\)

Example (3x3)

\(\begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix} - \begin{bmatrix}1 & 4 & 7\\2 & 5 & 8\\3 & 6 & 9\end{bmatrix} = \begin{bmatrix}0 & -2 & -4\\2 & 0 & -2\\4 & 2 & 0\end{bmatrix}\)

Properties
  • NOT Commutative: A − B ≠ B − A
  • Associative: (A − B) − C ≠ A − (B − C)
  • Identity: A − 0 = A (zero matrix)
  • Inverse operation: A − B = A + (−B)
  • Same dimensions: Result has same size as inputs


Description of Matrix Subtraction

Fundamentals

To subtract matrices, the matrices must match. That is, they must have the same number of rows and the same number of columns. This operation is related to matrix addition but produces different results.

General Formula (4×4):

4x4 Matrix subtraction formula 1 4x4 Matrix subtraction formula 2

How It Works

In order to perform a matrix subtraction, the corresponding matrix elements are subtracted. Each element in position (i,j) of matrix B is subtracted from the corresponding element at position (i,j) in matrix A.

  1. Check dimensions: Both matrices must be 4×4
  2. Subtract each element: Cij = Aij − Bij
  3. The result is a new matrix with the same dimensions

Detailed Example (3×3)

Step-by-Step Calculation

Matrix subtraction example 1 Matrix subtraction example 2

Complete Example:

Matrix subtraction example 3 Matrix subtraction example 4

Key Difference from Addition

Unlike matrix addition, matrix subtraction is NOT commutative:

Addition (Commutative):

A + B = B + A ✓

Subtraction (NOT Commutative):

A − B ≠ B − A ✗

Important: The order matters! A − B gives different results than B − A.

Practical Applications

Mathematics & Physics:

  • Computing differences between transformations
  • Finding error matrices in numerical methods
  • Calculating relative changes in systems
  • Image processing: difference images

Engineering & Computer Science:

  • Error analysis: Comparing actual vs. expected
  • Image differencing: Motion detection
  • Change detection: Before/after comparisons
  • Data analysis: Computing residuals
Relationship to Matrix Addition

Matrix subtraction can be expressed as addition with the negative matrix:

A − B = A + (−B)

Where −B means every element of B is multiplied by −1. This shows that subtraction is essentially addition with negated elements.

Properties of Matrix Subtraction

What Holds:

  • Same dimensions: Result is also 4×4
  • Zero matrix: A − 0 = A
  • Self-subtraction: A − A = 0
  • Distributive (scalar): k(A − B) = kA − kB

What Doesn't Hold:

  • NOT Commutative: A − B ≠ B − A
  • NOT Associative: (A − B) − C ≠ A − (B − C)
  • Order matters! Always subtract in correct order
  • Sign reversal: B − A = −(A − B)
Common Mistakes to Avoid
  • Wrong order: A − B is NOT the same as B − A. The result will have opposite signs: B − A = −(A − B).
  • Dimension mismatch: You cannot subtract a 3×3 matrix from a 4×4 matrix. Dimensions must match exactly.
  • Element-wise operation: Matrix subtraction is element-by-element, NOT like matrix multiplication which involves rows and columns.
  • Grouping matters: (A − B) − C ≠ A − (B − C). Always calculate left to right or use parentheses carefully.