A matrix denotes a number.

NCERT Exemplar · Q77

Q. A matrix denotes a number.

Telangana TsbieShort· 3mImportance★★★★★

87% · 158/182 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Matrix Notation

Matrix Notation: A First Look

You run a fruit stall selling apples, bananas, and oranges, and you track sales across Monday, Tuesday, and Wednesday. You could write three separate lists:

Monday: 10 apples, 5 bananas, 8 oranges
Tuesday: 7 apples, 12 bananas, 3 oranges
Wednesday: 9 apples, 6 bananas, 11 oranges

That works but is messy. A matrix is a cleaner way to arrange the same information — a rectangular grid of numbers organised into rows and columns.

The Intuition: A Table of Numbers

Think of a matrix as a spreadsheet:

Rows represent one category (a day of the week)
Columns represent another (a type of fruit)

For the stall:

\begin{bmatrix} 10 & 5 & 8 \\ 7 & 12 & 3 \\ 9 & 6 & 11 \end{bmatrix}

The first row is Monday's sales; the second column holds all banana sales across the three days: 5, 12, 6.

Note

A matrix is not a number — it's a collection of numbers in a specific shape. You can't say "this matrix equals 5" any more than "this table equals 5."

The Precise Statement

A matrix is a rectangular array of numbers arranged in $m$ rows and $n$ columns — an $m \times n$ matrix (read "m by n"). The numbers inside are its entries or elements. To name a specific entry we use two subscripts: row first, then column.

A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix}

Here $a_{ij}$ is the entry in row $i$ , column $j$ ; for example $a_{23}$ is second row, third column.

A = [a_{ij}]_{m \times n}

Shorthand for "the matrix $A$ whose entry in row $i$ , column $j$ is $a_{ij}$ , with $m$ rows and $n$ columns."

Key Vocabulary

Term	Meaning	Example
Order (size)	rows × columns	$3 \times 3$ for the fruit matrix
Square matrix	same number of rows and columns	$2 \times 2$ , $3 \times 3$
Row matrix	only one row	$[1 \quad 2 \quad 3]$ (a $1 \times 3$ matrix)
Column matrix	only one column	$\begin{bmatrix}4 \\ 5 \\ 6\end{bmatrix}$ (a $3 \times 1$ matrix)
Zero matrix	all entries are 0	$\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$

Start your 14-day free trial to unlock the full solution →

False: a matrix is a rectangular arrangement of numbers, not a single number.

Reading the statement

"A matrix denotes a number" claims a matrix is itself a scalar. That is not what a matrix is.

Why it is false

A matrix of order $m\times n$ is an ordered grid of $m\times n$ entries, for example

$\begin{bmatrix}2&-1\\0&3\end{bmatrix},$ …

Start your 14-day free trial to unlock the full solution →

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

TG EAPCET 2023Set eng-2023-05-14-AN1 markMCQ
$X_{4 \times 3}$
›Reveal solutionSolution
The key idea is to track matrix dimensions through transposition, multiplication, and inversion. The final order is $2 \times 2$ , so the correct option is (D).

We are given matrices with these dimensions:

$X_{4 \times 3}$ and $Y_{4 \times 3}$ (both have 4 rows, 3 columns)

$P_{2 \times 3}$ (2 rows, 3 columns)

We need the order of $\left[ P (X^T Y)^{-1} P^T \right]^T$ .

Concept and intuition

Matrix multiplication is only defined when the inner dimensions match. Transposition swaps rows and columns. Inversion is only possible for square matrices. So we must check that $(X^T Y)$ is square and invertible, then track dimensions step by step. The outer transpose flips the final result.

Step-by-step reasoning

Find $X^T$ :

$X$ is $4 \times 3$ , so $X^T$ is $3 \times 4$ .

Multiply $X^T Y$ :

$X^T$ is $3 \times 4$ , $Y$ is $4 \times 3$ .

Inner dimensions (4 and 4) match, so the product is $3 \times 3$ .

This is a square matrix, so its inverse exists (assuming it's invertible).

Invert $(X^T Y)$ :

$(X^T Y)^{-1}$ is also $3 \times 3$ .

Multiply $P$ by $(X^T Y)^{-1}$ :

$P$ is $2 \times 3$ , $(X^T Y)^{-1}$ is $3 \times 3$ .

Inner dimensions (3 and 3) match, so $P (X^T Y)^{-1}$ is $2 \times 3$ . …
Start your 14-day free trial to see the full solution →

←Previous questionQ76 — If A and B are symmetric matrices of same order, then AB is symmetric …→Next questionQ78 — Matrices of any order can be added.

Q.A matrix denotes a number.

Concept understanding — Matrix Notation

Matrix Notation: A First Look

The Intuition: A Table of Numbers

The Precise Statement

Key Vocabulary

Reading the statement

Why it is false

Method: Distinguish a matrix from a single number

Steps

Common Mistakes

Q.A matrix denotes a number.

Reading the statement

Why it is false

🎓Unlock everything free for 14 days

Unlock everything free for 14 days