If $A$ and $B$ are two matrices of the order $3 \times m$ and $3 \times n$, respectively, and $m = n$, then the order of matrix $(5A - 2B)$ is (A) $m \times 3$ (B) $3 \times 3$ (C) $m \times n$ (D) $3 \times n$

> [!TLDR] > Matrix subtraction requires both matrices to have the same order. Since $A$ is $3 \times m$ and $B$ is $3 \times n$, and $m = n$, both are $3 \times m$. The result $(5A - 2B)$ keeps the same order, so the answer is $3 \times m$ — which matches option **(D)** $3 \times n$. The key idea here is **matrix multiplication compatibility** — but actually, this problem is about **matrix addition and subtraction**, not multiplication. The rule is simpler: you can only add or subtract two matrices if they have exactly the same number of rows and the same number of columns. The result then has that same order. Let’s unpack what’s given. 1. **Identify the orders.** Matrix $A$ is $3 \times m$ — that’s 3 rows and $m$ columns. Matrix $B$ is $3 \times n$ — that’s 3 rows and $n$ columns. 2. **Use the condition $m = n$.** The problem tells us $m = n$. So $B$ is actually $3 \times m$ as well (since $n = m$). Both matrices now have the same order: $3 \times m$. 3. **Scalar multiplication doesn’t change the order.** Multiplying a matrix by a scalar (like $5$ or $-2$) just multiplies every entry — the shape stays exactly the same. So $5A$ is still $3 \times m$, and $2B$ is still $3 \times m$. 4. **Subtraction is element-wise.** Since $5A$ and $2B$ have the same order, we can subtract them entry by entry. The result $(5A - 2B)$ will also be a $3 \times m$ matrix. …

If A and B are two matrices of the order 3 × m and 3 × n, respectively, and m = n, then the order of matrix (5

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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 requires both matrices to have the same order. Since $A$ is $3 \times m$ and $B$ is $3 \times n$ , and $m = n$ , both are $3 \times m$ . The result $(5A - 2B)$ keeps the same order, so the answer is $3 \times m$ — which matches option (D) $3 \times n$ .

The key idea here is matrix multiplication compatibility — but actually, this problem is about matrix addition and subtraction, not multiplication. The rule is simpler: you can only add or subtract two matrices if they have exactly the same number of rows and the same number of columns. The result then has that same order.

Let’s unpack what’s given.

Identify the orders.

Matrix $A$ is $3 \times m$ — that’s 3 rows and $m$ columns.

Matrix $B$ is $3 \times n$ — that’s 3 rows and $n$ columns.
Use the condition $m = n$ .

The problem tells us $m = n$ . So $B$ is actually $3 \times m$ as well (since $n = m$ ). Both matrices now have the same order: $3 \times m$ .
Scalar multiplication doesn’t change the order.

Multiplying a matrix by a scalar (like $5$ or $-2$ ) just multiplies every entry — the shape stays exactly the same. So $5A$ is still $3 \times m$ , and $2B$ is still $3 \times m$ .
Subtraction is element-wise.

Since $5A$ and $2B$ have the same order, we can subtract them entry by entry. The result $(5A - 2B)$ will also be a $3 \times m$ matrix. …

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Q.If $A$ and $B$ are two matrices of the order $3 \times m$ and $3 \times n$ , respectively, and $m = n$ , then the order of matrix $(5A - 2B)$ is
(A) $m \times 3$
(B) $3 \times 3$
(C) $m \times n$
(D) $3 \times n$

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: Order of a linear combination of matrices

Steps

Common Mistakes

Q.If AAA and BBB are two matrices of the order 3×m3 \times m3×m and 3×n3 \times n3×n, respectively, and m=nm = nm=n, then the order of matrix (5A−2B)(5A - 2B)(5A−2B) is (A) m×3m \times 3m×3 (B) 3×33 \times 33×3 (C) m×nm \times nm×n (D) 3×n3 \times n3×n

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Q.If $A$ and $B$ are two matrices of the order $3 \times m$ and $3 \times n$ , respectively, and $m = n$ , then the order of matrix $(5A - 2B)$ is
(A) $m \times 3$
(B) $3 \times 3$
(C) $m \times n$
(D) $3 \times n$

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