Assume $X$, $Y$, $Z$, $W$ and $P$ are matrices of order $2 \times n$, $3 \times k$, $2 \times p$, $n \times 3$ and $p \times k$, respectively. If $n = p$, then the order of the matrix $7X - 5Z$ is: (A) $p \times 2$ (B) $2 \times n$ (C) $n \times 3$ (D) $p \times n$

Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively. If n = p, th

Exercise 3.2 · Q22

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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 subtraction is only defined when both matrices have the same order. Since $X$ is $2 \times n$ and $Z$ is $2 \times p$ , and we are given $n = p$ , both are $2 \times n$ . So $7X - 5Z$ is also $2 \times n$ , which matches option (B).

The key idea here is matrix compatibility for addition and subtraction. You can only add or subtract two matrices if they have exactly the same number of rows and the same number of columns. Multiplying a matrix by a scalar (like 7 or 5) does not change its order — it just scales every entry.

Let’s lay out what we know:

Order of $X$ : $2 \times n$

So $X$ has 2 rows and $n$ columns.
Order of $Z$ : $2 \times p$

So $Z$ has 2 rows and $p$ columns.
Given condition: $n = p$

This means the number of columns in $X$ equals the number of columns in $Z$ .
Effect of scalar multiplication:

$7X$ still has order $2 \times n$ , and $5Z$ still has order $2 \times p$ . Since $n = p$ , both $7X$ and $5Z$ are $2 \times n$ .
Subtraction is defined:

Because both matrices now have the same order ( $2 \times n$ ), $7X - 5Z$ is defined and its order remains $2 \times n$ . …

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←Previous question · 3.4.6Q21 — Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n ×…→Next section · 3.5Transpose of a Matrix

Q.Assume $X$ , $Y$ , $Z$ , $W$ and $P$ are matrices of order $2 \times n$ , $3 \times k$ , $2 \times p$ , $n \times 3$ and $p \times k$ , respectively. If $n = p$ , then the order of the matrix $7X - 5Z$ is: (A) $p \times 2$ (B) $2 \times n$ (C) $n \times 3$ (D) $p \times n$

Concept understanding — Matrix Addition

Matrix Addition: The Intuition

The Precise Statement

The One Rule You Cannot Break

Properties That Feel Natural

A Quick Check …

Method: Finding the order of a scalar combination of matrices

Steps

Common Mistakes

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