CBSE Sample Paper 2024-25 · Section A · 1 mark

Q2: Restrictions on $n, k, p$ so that $PY + WY$ is defined

Sample 2024-25Ch 3 · Matrices

The question

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. Then the restriction on $n, k$ and $p$ so that $PY + WY$ will be defined are:

(A) $k = 3, p = n$

(B) $k$ is arbitrary, $p = 2$

(D) $k = 2, p = 3$

Exam answer

$P_{p \times k}\,Y_{3 \times k}$ is defined only if $k = 3$ ⁠. Then $PY$ is of order $p \times k$ ⁠.

Why?

A product $XY$ is defined only when the number of columns of $X$ equals the number of rows of $Y$ ⁠. The answer then has the rows of $X$ and the columns of $Y$ ⁠:

X_{m \times n}\,Y_{n \times r} = (XY)_{m \times r}

$P$ has $k$ columns and $Y$ has 3 rows, so we need $k = 3$ ⁠.

$W_{n \times 3}\,Y_{3 \times k}$ is defined and is of order $n \times k$ ⁠.

Why?

$W$ has 3 columns and $Y$ has 3 rows, so $WY$ is always defined. It takes its $n$ rows from $W$ and its $k$ columns from $Y$ ⁠.

$PY + WY$ is defined only if both have the same order: $p \times k = n \times k$ ⁠, so $p = n$ ⁠.

Why?

Two matrices can be added only when they have the same order. The columns already match ( $k$ and $k$ ⁠), so the rows must match: $p = n$ ⁠.

Option (C) misses $p = n$ ⁠, option (B) fixes $p = 2$ instead, and in option (D) $k = 2$ makes $PY$ undefined.

$\therefore\ k = 3$ and $p = n$ ⁠.

Answer: (A) $k = 3, p = n$

Where this comes from

Chapter 3, Matrices: Order of matrices: when PY + WY is defined.

Question type: One-mark MCQs

How to spot it: click to see

Type 6. One-mark MCQs: orders and definitions

Spot it: four options, asking when a sum or product is defined, what order it has, or what a word such as symmetric or inverse means.

Here: Write the orders: $P$ is $p \times k$ and $Y$ is $3 \times k$ ⁠, so $PY$ needs $k = 3$ ⁠. $PY$ is $p \times k$ and $WY$ is $n \times k$ ⁠, so the sum needs $p = n$ ⁠.

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Q2: Restrictions on n,k,pn, k, pn,k,p so that PY+WYPY + WYPY+WY is defined

Q2: Restrictions on $n, k, p$ so that $PY + WY$ is defined