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. The restriction on $n$, $k$ and $p$ so that $PY + WY$ will be defined are: (A) $k = 3,\ p = n$ (B) $k$ is arbitrary, $p = 2$ (C) $p$ is arbitrary, $k = 3$ (D) $k = 2,\ p = 3$

Concept: Matrix multiplication is defined only when the number of columns in the first matrix equals the number of rows in the second matrix (compatibility condition).

Step 1: Check $PY$

$P$ is $p \times k$, $Y$ is $3 \times k$. For $PY$ to be defined, $k$ (columns of $P$) must equal $3$ (rows of $Y$). So $k = 3$.

Step 2: Check $WY$

$W$ is $n \times 3$, $Y$ is $3 \times k$. For $WY$ to be defined, $3$ (columns of $W$) must equal $3$ (rows of $Y$) — this is already satisfied. The product $WY$ exists for any $n$ and $k$, but since $k=3$ from Step 1, it is fine. …

Two rules in play

• Multiplication: a product $MN$ is defined only when the number of columns of $M$ equals the number of rows of $N$.
• Addition: two matrices can be added only when they have the same order.

Given orders:

• $P$: $p \times k$
• $Y$: $3 \times k$
• $W$: $n \times 3$

Condition 1 — $PY$ is defined

$P$ is $p \times k$ and $Y$ is $3 \times k$. Multiplication requires columns of $P$ ($=k$) to equal rows of $Y$ ($=3$), so

Then $PY$ has order $p \times k = p \times 3$.

Condition 2 — $WY$ is defined …

Method: Finding restrictions so a matrix expression is defined

For an expression that combines products and sums, apply two rules in turn: each product needs matching inner dimensions, and any sum needs both terms to have the same order.

Steps

Step 1: Impose the multiplication condition on each product

For $PY$ with $P$ of order $p\times k$ and $Y$ of order $3\times k$, the columns of $P$ must equal the rows of $Y$: $k=3$.

Step 2: Determine each product's resulting order …

Common Mistakes

Mistake 1: Checking only that the products exist and forgetting the sum's order-match

Why it's wrong: even if $PY$ and $WY$ are each defined, adding them requires equal orders, which gives the extra condition $p=n$. Correct approach: apply the addition rule after the multiplication rule.

Mistake 2: Confusing which dimension must match for multiplication …