Q.If for three matrices 𝐴 = [𝑎𝑖𝑗]𝑚×4 , B = [𝑏𝑖𝑗]𝑛×3 𝑎𝑛𝑑 C = [𝑐𝑖𝑗]𝑝×𝑞 products 𝐴𝐵 and 𝐴𝐶 both are defined and are square matrices of same order, then value of 𝑚, 𝑛, 𝑝 and 𝑞 are:
(A) 𝑚 = 𝑞 = 3 𝑎𝑛𝑑 𝑛 = 𝑝 = 4
(B) 𝑚 = 2, 𝑞 = 3 𝑎𝑛𝑑 𝑛 = 𝑝 = 4
(C) 𝑚 = 𝑞 = 4 𝑎𝑛𝑑 𝑛 = 𝑝 = 3
(D) 𝑚 = 4, 𝑝 = 2 𝑎𝑛𝑑 𝑛 = 𝑞 = 3

Apply the inner-dimension rule for products, then the square condition on the outer dimensions.

Step 1 — $AB$ defined. $A$ is $m\times 4$, $B$ is $n\times 3$. Columns of $A$ must equal rows of $B$: $4=n$, so $n=4$.

Step 2 — $AB$ square. $AB$ has order $m\times 3$. Square means rows $=$ columns: $m=3$.

Step 3 — $AC$ defined. $A$ is $m\times 4$, $C$ is $p\times q$. Columns of $A$ must equal rows of $C$: $4=p$, so $p=4$. …

The two rules we need

For a product $XY$ to exist, the number of columns of $X$ must equal the number of rows of $Y$, and the result takes the outer dimensions. A matrix is square when its number of rows equals its number of columns.

We are told $A=[a_{ij}]_{m\times 4}$, $B=[b_{ij}]_{n\times 3}$, $C=[c_{ij}]_{p\times q}$, and that both $AB$ and $AC$ are defined square matrices of the same order.

Working the conditions

1. $AB$ is defined. Columns of $A$ (which is $4$) must equal rows of $B$ (which is $n$):

2. $AB$ is square. With $A_{m\times 4}$ and $B_{4\times 3}$, the product $AB$ has order $m\times 3$. For a square matrix the two must be equal:

So $AB$ is $3\times 3$.

3. $AC$ is defined. Columns of $A$ ($4$) must equal rows of $C$ ($p$): $4=p\ \Rightarrow\ p=4.$ …

Method: Deducing unknown orders from "defined" and "square" conditions

Use this whenever a problem gives matrices with unknown orders and tells you certain products are defined and/or square, and asks you to pin down the unknowns.

Steps

Step 1: Turn each "product is defined" into an inner-dimension equation.

For a product $XY$ to exist, the columns of $X$ must equal the rows of $Y$. Write that equality for every product the problem says is defined.

Step 2: Turn each "product is square" into an outer-dimension equation. …

Common Mistakes

Mistake 1: Assuming a square product must come from square factors.

Why it's wrong: order comes from the outer dimensions, so $A_{3\times4}B_{4\times3}$ is a $3\times3$ (square) product even though neither factor is square. Correct approach: read "square" off the product's own order $m\times q$, not the factors.

Mistake 2: Matching outer dimensions for "defined". …