Matrix multiplication is _____ over addition.

NCERT Exemplar · Q69

Q. Matrix multiplication is _________ over addition.

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Concept understanding — Matrix Addition

Matrix Addition: The Intuition

You run a fruit stall and record apples and bananas sold each morning and afternoon in a table:

Time	Apples	Bananas
Morning	10	5
Afternoon	8	12

That's a matrix — a rectangular array of numbers. Your friend's stall has its own table for the same day (morning: 6 apples, 7 bananas; afternoon: 4 apples, 9 bananas). To get the combined sales, you add the numbers in the same position: morning apples with morning apples, afternoon bananas with afternoon bananas, and so on.

That's matrix addition: you add corresponding entries — numbers in the same row and column.

The Precise Statement

(A + B)_{ij} = A_{ij} + B_{ij}

If $A$ and $B$ have the same size (same number of rows and columns), their sum $A + B$ is a matrix of that size where each entry is the sum of the corresponding entries.

Example:

A = \begin{bmatrix} 2 & 5 \\ 1 & -3 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 4 \\ 7 & 2 \end{bmatrix}

A + B = \begin{bmatrix} 2+0 & 5+4 \\ 1+7 & -3+2 \end{bmatrix} = \begin{bmatrix} 2 & 9 \\ 8 & -1 \end{bmatrix}

The One Rule You Cannot Break

Watch out

You can only add matrices with the exact same dimensions. A $2 \times 3$ matrix cannot be added to a $3 \times 2$ matrix — the positions don't match.

Properties That Feel Natural

Matrix addition behaves like ordinary number addition, inheriting these from the addition of individual entries:

Commutative: $A + B = B + A$
Associative: $(A + B) + C = A + (B + C)$
Zero matrix: there's a matrix $O$ (all zeros) with $A + O = A$

A Quick Check …

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Matrix multiplication is distributive over addition. This means for matrices $A$ , $B$ , $C$ of compatible sizes, we have $A(B+C) = AB + AC$ and $(B+C)A = BA + CA$ .

Why This Property Matters

Matrix multiplication and addition are the two fundamental operations you'll use constantly in linear algebra. The question asks about their relationship — specifically, what happens when you multiply a matrix by a sum of two others.

Think of it like arithmetic with numbers: $3 \times (4 + 5) = 3 \times 4 + 3 \times 5$ . That's distributivity. For matrices, the same idea holds, but with a crucial twist: order matters because matrix multiplication is not commutative. So we actually get two distributive laws — one for left-multiplication and one for right-multiplication.

Watch out

A common mistake is to assume $A(B+C) = AB + AC$ works for any three matrices. It only works when the sizes are compatible: $A$ must be $m \times n$ , and both $B$ and $C$ must be $n \times p$ (so $B+C$ is defined, and $A$ can multiply it). Always check dimensions first.

Step-by-Step Verification

Let's prove the left distributive law: $A(B+C) = AB + AC$ .

1. Set up the matrices.

Let $A$ be an $m \times n$ matrix, and $B$ and $C$ be $n \times p$ matrices. Then $B+C$ is also $n \times p$ , so $A(B+C)$ is $m \times p$ — same as $AB$ and $AC$ , which are also $m \times p$ . The dimensions match, so the equality is at least possible.

2. Write the $(i,j)$ entry of $A(B+C)$ .

The $(i,j)$ entry of a product is the dot product of row $i$ of the first matrix with column $j$ of the second. So:

[A(B+C)]_{ij} = \sum_{k=1}^{n} A_{ik} (B+C)_{kj}

3. Use the definition of matrix addition.

The $(k,j)$ entry of $B+C$ is simply $B_{kj} + C_{kj}$ . So:

[A(B+C)]_{ij} = \sum_{k=1}^{n} A_{ik} (B_{kj} + C_{kj})

4. Distribute the sum over the addition inside.

This is just algebra with real numbers (the entries are numbers):

[A(B+C)]_{ij} = \sum_{k=1}^{n} (A_{ik} B_{kj} + A_{ik} C_{kj})

5. Split the sum into two separate sums.

A sum of a sum is the sum of the individual sums: …

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Q.Matrix multiplication is _________ over addition.

Concept understanding — Matrix Addition

Matrix Addition: The Intuition

The Precise Statement

The One Rule You Cannot Break

Properties That Feel Natural

A Quick Check …

Why This Property Matters

Step-by-Step Verification

Method: Recognise how multiplication interacts with addition

Steps

Q.Matrix multiplication is _________ over addition.

Why This Property Matters

Step-by-Step Verification

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