If $A = [a_{ij}]$ is an identity matrix, then which of the following is true? (A) $a_{ij} = \begin{cases} 0, & \text{if } i = j \\ 1, & \text{if } i \neq j \end{cases}$ (B) $a_{ij} = 1,\ \forall\, i, j$ (C) $a_{ij} = 0,\ \forall\, i, j$ (D) $a_{ij} = \begin{cases} 0, & \text{if } i \neq j \\ 1, & \text{if } i = j \end{cases}$

> [!TLDR] > An identity matrix has 1’s on the main diagonal and 0’s everywhere else. The correct description is option **(D)**. The definition of an identity matrix is one of the first things you learn in matrices — and it’s also one of the easiest to mix up if you’re not careful. The key idea is simple: an identity matrix acts like the number 1 in multiplication. When you multiply any matrix by the identity (of the right size), you get the original matrix back. For that to work, the identity must have 1’s only where the row and column numbers are the same (the diagonal), and 0’s everywhere else. Let’s walk through the options one by one. 1. **Option (A)** says: $a_{ij} = \begin{cases} 0, & \text{if } i = j \\ 1, & \text{if } i \neq j \end{cases}$ This is the exact opposite of what we need — it puts 0’s on the diagonal and 1’s off it. That matrix is called a *counter-identity* or sometimes a *J* matrix (all ones except zeros on the diagonal). It definitely does not behave like the multiplicative identity. So (A) is wrong. 2. **Option (B)** says: $a_{ij} = 1,\ \forall\, i, j$ This is a matrix of all 1’s. Multiply any vector by this and you’ll get a vector of sums — not the original vector. So (B) is also wrong. 3. **Option (C)** says: $a_{ij} = 0,\ \forall\, i, j$ This is the zero matrix. Multiplying by the zero matrix gives zero, not the original matrix. So (C) is clearly wrong. 4. **Option (D)** says: $a_{ij} = \begin{cases} 0, & \text{if } i \neq j \\ 1, & \text{if } i = j \end{cases}$ This is exactly the definition: 1 on the diagonal (where $i = j$) and 0 elsewhere. This is the identity matrix, often denoted $I_n$ for an $n \times n$ matrix. > [!WARNING] > A common mistake is to reverse the conditions — putting 1 where $i \neq j$ and 0 where $i = j$. That gives you a matrix that is *not* the identity, and it won’t work as a multiplicative identity. > [!TIP] > A quick way to remember: the identity matrix looks like a

If A = [aᵢⱼ] is an identity matrix, then which of the following is true? (A) aᵢⱼ = 0, if i = j \1, if i ≠ j (B

Question

A = [a_{ij}]

CBSECBSE Class XII Board 2024MCQ· 1mImportance★★★★★

✓ Free question

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	$\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$

Watch out

A common mistake: mixing up rows and columns. Always row first, then column — think of a row as horizontal (a row of seats) and a column as vertical (a column in a building).

Why Bother?

Matrices let us:

Compactify large amounts of data (imagine 30 days and 10 fruits)
Perform operations on entire datasets at once (adding two weeks of sales, scaling prices)
Solve systems of equations efficiently

For now, get comfortable with the notation. When you see $\begin{bmatrix}2 & -1 \\ 0 & 3\end{bmatrix}$ , think: "a $2 \times 2$ square matrix; the entry in row 2, column 1 is 0." Matrix notation — the brackets, the subscripts, the order — is simply the language for these rectangular arrays, and mastering it makes addition, multiplication, and determinants much easier to follow.

Matrix notation — rows, columns, order, and the aᵢⱼ indexing convention — is the very first idea introduced in the CBSE Class 12 Matrices chapter, making "matrix definition and notation class 12" a frequently searched starting point for this unit. A solid grip on this notation is what makes every later topic in the chapter, from addition to determinants, easier to follow.

An identity matrix has 1’s on the main diagonal and 0’s everywhere else. The correct description is option (D).

The definition of an identity matrix is one of the first things you learn in matrices — and it’s also one of the easiest to mix up if you’re not careful. The key idea is simple: an identity matrix acts like the number 1 in multiplication. When you multiply any matrix by the identity (of the right size), you get the original matrix back. For that to work, the identity must have 1’s only where the row and column numbers are the same (the diagonal), and 0’s everywhere else.

Let’s walk through the options one by one.

Option (A) says:

$a_{ij} = \begin{cases} 0, & \text{if } i = j \\ 1, & \text{if } i \neq j \end{cases}$

This is the exact opposite of what we need — it puts 0’s on the diagonal and 1’s off it. That matrix is called a counter-identity or sometimes a J matrix (all ones except zeros on the diagonal). It definitely does not behave like the multiplicative identity. So (A) is wrong.
Option (B) says:

$a_{ij} = 1,\ \forall\, i, j$

This is a matrix of all 1’s. Multiply any vector by this and you’ll get a vector of sums — not the original vector. So (B) is also wrong.
Option (C) says:

$a_{ij} = 0,\ \forall\, i, j$

This is the zero matrix. Multiplying by the zero matrix gives zero, not the original matrix. So (C) is clearly wrong.
Option (D) says:

$a_{ij} = \begin{cases} 0, & \text{if } i \neq j \\ 1, & \text{if } i = j \end{cases}$

This is exactly the definition: 1 on the diagonal (where $i = j$ ) and 0 elsewhere. This is the identity matrix, often denoted $I_n$ for an $n \times n$ matrix.

Watch out

A common mistake is to reverse the conditions — putting 1 where $i \neq j$ and 0 where $i = j$ . That gives you a matrix that is not the identity, and it won’t work as a multiplicative identity.

Tip

A quick way to remember: the identity matrix looks like a diagonal of 1’s. The word “diagonal” is your clue — only where row = column.

✓Final answer

The correct option is (D) .

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

CBSE 2026Set ANNUAL1 markMCQ
$Q. The order of the matrix A = [[2, 5, 19, -7], [35, -2, 5, 12]] is:$
(a)
(i) 4x2
(b)
(ii) 8
(c)
(iii) 2x3
(d)
(iv) 2x4
›Reveal solutionSolution
The matrix has 2 rows and 4 columns, so its order is $2\times 4$ — option (iv).

Concept. For a matrix, the order is written as $(\text{rows})\times(\text{columns})$ .

Steps.

$A=\begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & 5 & 12 \end{bmatrix}$ .

Count the horizontal lines (rows): there are $2$ .

Count the entries in each row (columns): there are $4$ .

Hence the order is $2\times 4$ .

✓Final answer
$= 2\times 4$
CBSE 2026Set ANNUAL1 mark
$Q. Fill in the blank: A matrix is said to be a column matrix if it has only ______.$
›Reveal solutionSolution

A column matrix has a single column.

A matrix of order $m\times1$ (one column, any number of rows) is called a column matrix.

✓Final answer
$Only one column.$
CBSE 2025Set A1 mark
$Q. Write definition of scalar matrix.$
›Reveal solutionSolution
Scalar matrix = diagonal matrix with all equal diagonal entries.

A square matrix $A = [a_{ij}]$ is called a scalar matrix if:

It is a diagonal matrix, i.e. $a_{ij} = 0$ for all $i\ne j$ (all off-diagonal entries are zero), AND

All the diagonal entries are equal, i.e. $a_{11}=a_{22}=\cdots=a_{nn}=k$ for some fixed scalar $k$ .

Equivalently, $A = kI$ , where $I$ is the identity matrix of the same order. For example, $\begin{bmatrix}5&0\\0&5\end{bmatrix}$ is a scalar matrix.

✓Final answer
$A=kI$
CBSE 2025Set ANNUAL1 markMCQ
$A=[a_{ij}]_{m\times n}$
(a) $m>1$
(b) $m=1$
(c) $n>1$
(d) $n=1$
›Reveal solutionSolution

A matrix $A=[a_{ij}]_{m\times n}$ is called a column matrix when it has only one column, i.e. $n=1$ .

By definition, a column matrix (or column vector) has $m$ rows but a single column, so its order is $m\times 1$ . This means the number of columns $n$ must equal $1$ , regardless of $m$ .

✓Final answer
$n=1$
CBSE 2024Set 65/3/11 markMCQ
$A = [a_{ij}]$
›Reveal solutionSolution
An identity matrix has 1’s on the main diagonal and 0’s everywhere else. The correct description is option (D).

The definition of an identity matrix is one of the first things you learn in matrices — and it’s also one of the easiest to mix up if you’re not careful. The key idea is simple: an identity matrix acts like the number 1 in multiplication. When you multiply any matrix by the identity (of the right size), you get the original matrix back. For that to work, the identity must have 1’s only where the row and column numbers are the same (the diagonal), and 0’s everywhere else.

Let’s walk through the options one by one.

Option (A) says:

$a_{ij} = \begin{cases} 0, & \text{if } i = j \\ 1, & \text{if } i \neq j \end{cases}$

This is the exact opposite of what we need — it puts 0’s on the diagonal and 1’s off it. That matrix is called a counter-identity or sometimes a J matrix (all ones except zeros on the diagonal). It definitely does not behave like the multiplicative identity. So (A) is wrong.

Option (B) says:

$a_{ij} = 1,\ \forall\, i, j$

This is a matrix of all 1’s. Multiply any vector by this and you’ll get a vector of sums — not the original vector. So (B) is also wrong.

Option (C) says:

$a_{ij} = 0,\ \forall\, i, j$

This is the zero matrix. Multiplying by the zero matrix gives zero, not the original matrix. So (C) is clearly wrong.

Option (D) says:

$a_{ij} = \begin{cases} 0, & \text{if } i \neq j \\ 1, & \text{if } i = j \end{cases}$

This is exactly the definition: 1 on the diagonal (where $i = j$ ) and 0 elsewhere. This is the identity matrix, often denoted $I_n$ for an $n \times n$ matrix.

Watch out

A common mistake is to reverse the conditions — putting 1 where $i \neq j$ and 0 where $i = j$ . That gives you a matrix that is not the identity, and it won’t work as a multiplicative identity.

Tip

A quick way to remember: the identity matrix looks like a diagonal of 1’s. The word “diagonal” is your clue — only where row = column.

✓Final answer
$The correct option is (D) .$
CBSE 2023Set A1 mark
$Q. What is column matrix?$
›Reveal solutionSolution

A column matrix has exactly one column and any number of rows — order $m\times1$ .

A matrix $A=[a_{ij}]_{m\times n}$ is called a column matrix when $n=1$ , i.e. it has only a single column: $A=\begin{bmatrix}a_{11}\\a_{21}\\\vdots\\a_{m1}\end{bmatrix}$ . For example, $\begin{bmatrix}2\\-1\\5\end{bmatrix}$ is a $3\times1$ column matrix.

✓Final answer
$m\times1$
CBSE 2022Set ANNUAL1 mark
$3 \times 3$
›Reveal solutionSolution

An identity matrix has 1's on the main diagonal and 0's everywhere else.

The $3\times 3$ identity matrix is $I_3=\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$ .

✓Final answer
$I_3=\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$
CBSE 2019Set ANNUAL1 markMCQ
$3 \times 3$
(a) $\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}$
(b) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
(c) $\begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}$
(d) $\begin{bmatrix} 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{bmatrix}$
›Reveal solutionSolution

The $3\times3$ identity matrix is $\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$ .

A unit (identity) matrix $I$ is a square matrix whose diagonal entries are all $1$ and every off-diagonal entry is $0$ . It satisfies $AI=IA=A$ . Only option (b) has this structure.

✓Final answer
$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$

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