lesson
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When you add 3+5, you get the same answer as 5+3. But if you pack thousands of numbers into multi-dimensional grids, does that flexibility survive, or do grids lock numbers in place?
A matrix is a rectangular grid of numbers organized into rows and columns, where you add two matrices simply by adding their corresponding entries—the numbers in the exact same row and column position.
In 1858, British mathematician Arthur Cayley formalized matrix algebra to solve systems of linear equations, discovering that matrix addition inherits the exact same algebraic foundations as regular arithmetic.
📊An interactive-style visual diagram showing entry-by-entry matrix addition. On the left, Matrix A [2, 5; -1, 4] and Matrix B [3, 1; 7, -2] are displayed with rounded brackets. Color-coded dashed arrows link matching cell positions: top-left (2 + 3 = 5, in blue #2563eb), top-right (5 + 1 = 6, in teal #0d9488), bottom-left (-1 + 7 = 6, in purple #7c3aed), and bottom-right (4 + (-2) = 2, in amber #d97706). On the right, the resulting Matrix (A + B) [5, 6; 6, 2] is shown with matching colored cells. Card background #ffffff, border #e2e8f0, text #1e293b, responsive layout.
What happens if we flip the matrices and add B+A instead of A+B?
The Commutative Property of Addition
The commutative property states that changing the order of the terms being added does not change the total sum. For any two matrices A and B with the exact same dimensions, A+B=B+A.
Let's test this with A=[40−27] and B=[1−583] to see both directions in action.
📊A side-by-side comparison diagram verifying commutativity. Top half shows 'A + B': [4, -2; 0, 7] + [1, 8; -5, 3] = [4+1, -2+8; 0+(-5), 7+3] = [5, 6; -5, 10]. Bottom half shows 'B + A': [1, 8; -5, 3] + [4, -2; 0, 7] = [1+4, 8+(-2); -5+0, 3+7] = [5, 6; -5, 10]. An equal sign badge with a green checkmark connects the two final matrices with the label 'A + B = B + A'. Dark blue accents #1e3a8a, emerald green check badge #059669, soft gray container background #f8fafc, crisp borders #cbd5e1.
Why does this always hold true? Each position in the sum matrix is formed by adding individual real numbers, and because regular number addition is commutative (aij+bij=bij+aij), the entire matrix sum must also commute.
Adding two matrices in any order gives the same result—but what happens when we group three matrices using parentheses?
The Associative Property of Addition
The associative property states that regrouping terms does not change the final sum: (A+B)+C=A+(B+C) for any matrices of the same dimensions.