Construct a $2 \times 2$ matrix where (i) $a_{ij} = \dfrac{(i-2j)^2}{2}$ (ii) $a_{ij} = |-2i+3j|$

Construct a 2 × 2 matrix where (i) aᵢⱼ = ((i-2j)²)/(2) (ii) aᵢⱼ = |-2i+3j|

NCERT Exemplar · Q3

2 \times 2

(i)

a_{ij} = \dfrac{(i-2j)^2}{2}

(ii)

a_{ij} = |-2i+3j|

Uttarakhand UbseShort· 2mImportance★★★★★

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Concept understanding — Matrix Construction

Matrix Construction: Building a Grid of Numbers

A teacher recording attendance for $30$ students over $5$ days could keep separate lists — but that is messy. Instead, draw a grid: rows for students, columns for days, each cell a $1$ (present) or $0$ (absent). That grid is a matrix. Constructing a matrix means deciding its shape and what number sits in each cell.

Why a Grid?

Every cell of a matrix has a unique address $(i, j)$ — row $i$ , column $j$ — so the entry in row $2$ , column $3$ is written $a_{23}$ . A grid beats a plain list because so many problems have two natural dimensions: a system of equations (equation $\times$ variable), a digital image (row $\times$ column of pixels), or a network (source node $\times$ destination node). The grid lets operations act on both dimensions at once.

The Precise Form

A matrix $A$ of order $m \times n$ (" $m$ by $n$ ") has $m$ rows and $n$ columns:

A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}, \qquad A = [a_{ij}]_{m \times n}.

Each $a_{ij}$ is an entry: the first index $i$ is the row, the second $j$ is the column.

How You Construct One

To build a matrix you specify:

Dimensions — how many rows $m$ and columns $n$ .
Entry rule — what number fills each cell: an explicit list, a formula in $i$ and $j$ , or data from a problem.
Placement — order matters; swapping rows or columns gives a different matrix.

Explicit: a $2 \times 3$ matrix with rows $(1, 0, -2)$ and $(3, 5, 7)$ is

A = \begin{pmatrix} 1 & 0 & -2 \\ 3 & 5 & 7 \end{pmatrix}.

Formula-based: for a $3 \times 3$ matrix with $a_{ij} = i^2 - j$ , we get $a_{11} = 0$ , $a_{12} = -1$ , $a_{21} = 3$ , giving

A = \begin{pmatrix} 0 & -1 & -2 \\ 3 & 2 & 1 \\ 8 & 7 & 6 \end{pmatrix}.

Watch out

Do not confuse $a_{ij}$ with $a_{ji}$ . The first index is always the row, the second the column — so $a_{23}$ is row $2$ , column $3$ .

Important

A matrix is not just a set of numbers — it is an ordered arrangement. The same numbers placed differently give a different matrix. When a problem says "construct $A = [a_{ij}]$ where $a_{ij} = \dots$ ", fix the dimensions first, then fill the cells one by one using the rule.

Constructing a matrix from a given formula for its entries, such as aᵢⱼ = i² − j, is a standard NCERT exercise type in the CBSE Class 12 Matrices chapter, and "construct a 3x3 matrix whose elements are given by formula" is a frequently searched question format. This skill is regularly tested in board exams as a straightforward, formula-substitution-based question.

Evaluate each formula at $(i,j)=(1,1),(1,2),(2,1),(2,2)$ . Part (i): $\begin{pmatrix} \tfrac{1}{2} & \tfrac{9}{2} \\0 & 2 \end{pmatrix}$ . Part (ii): $\begin{pmatrix} 1 & 4 \\1 & 2 \end{pmatrix}$ .

A $2\times 2$ matrix has entries $a_{ij}$ where $i$ is the row ( $1,2$ ) and $j$ is the column ( $1,2$ ). We simply substitute each $(i,j)$ into the given rule.

Part (i): $a_{ij}=\dfrac{(i-2j)^2}{2}$

a_{11}=\frac{(1-2)^2}{2}=\frac{1}{2},\qquad a_{12}=\frac{(1-4)^2}{2}=\frac{9}{2},

a_{21}=\frac{(2-2)^2}{2}=0,\qquad a_{22}=\frac{(2-4)^2}{2}=\frac{4}{2}=2.

A=\begin{pmatrix} \tfrac{1}{2} & \tfrac{9}{2} \\[2pt] 0 & 2 \end{pmatrix}.

Part (ii): $a_{ij}=|-2i+3j|$

a_{11}=|-2+3|=1,\qquad a_{12}=|-2+6|=4,

a_{21}=|-4+3|=|-1|=1,\qquad a_{22}=|-4+6|=2.

A=\begin{pmatrix} 1 & 4 \\1 & 2 \end{pmatrix}.

✓Final answer

\begin{pmatrix} \tfrac{1}{2} & \tfrac{9}{2} \\[2pt] 0 & 2 \end{pmatrix}

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Q.Construct a $2 \times 2$ matrix where

Concept understanding — Matrix Construction

Matrix Construction: Building a Grid of Numbers

Why a Grid?

The Precise Form

How You Construct One

Part (i): $a_{ij}=\dfrac{(i-2j)^2}{2}$

Part (ii): $a_{ij}=|-2i+3j|$

Method: Constructing a matrix from a formula $a_{ij}=f(i,j)$

Steps

Common Mistakes

Q.Construct a 2×22 \times 22×2 matrix where

Part (i): aij=(i−2j)22a_{ij}=\dfrac{(i-2j)^2}{2}aij​=2(i−2j)2​

Part (ii): aij=∣−2i+3j∣a_{ij}=|-2i+3j|aij​=∣−2i+3j∣

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Q.Construct a $2 \times 2$ matrix where

Part (i): $a_{ij}=\dfrac{(i-2j)^2}{2}$

Part (ii): $a_{ij}=|-2i+3j|$

Unlock everything free for 14 days