A = [a_{ij}]_{m×n} is a square matrix, if- (a) (i) m n (d) (iv) m^2 = n

A = [aᵢⱼ]m×n is a square matrix, if- (a) (i) m < n – Class 12 Mathematics

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Q.A = [a_{ij}]_{m×n} is a square matrix, if-

(a)

(i) m < n

(b)

(ii) m = n

(c)

(iii) m > n

(d)

(iv) m^2 = n

Uttarakhand UbseUttarakhand Board Intermediate (Class 12) 2019MCQ· 1mImportance★★★★★

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Concept understanding — Square Matrix Definition

What is a Square Matrix? The Intuition First

Arrange numbers in a grid. If the grid has the same number of rows as columns — like a chessboard (8×8) or a tic-tac-toe board (3×3) — that's a square matrix. The name comes from the shape: it looks like a square, not a rectangle.

Why care? Square matrices are the only ones that can be multiplied by themselves, have a determinant, or be inverted — the workhorses behind solving systems of equations, computer graphics, and quantum mechanics.

The Precise Definition

A = [a_{ij}]_{m \times n} \text{ is a square matrix if and only if } m = n

That is, the number of rows equals the number of columns. If a matrix has $m$ rows and $n$ columns its order is $m \times n$ ; for a square matrix we write the order as $n \times n$ , or say it's "of order $n$ ".

Example: $B = \begin{bmatrix} 2 & -1 & 0 \\ 3 & 5 & 7 \\ -4 & 1 & 8 \end{bmatrix}$ is a $3 \times 3$ square matrix.

Counterexample: $C = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$ is $3 \times 2$ — 3 rows, 2 columns. Not square, but rectangular.

Key Properties You'll Meet

Once a matrix is square, several features become possible:

Main diagonal: the entries $a_{11}, a_{22}, \dots, a_{nn}$ (top-left to bottom-right).
Determinant: a single number $\det(A)$ (or $|A|$ ); rectangular matrices don't have one.
Inverse: a square $A$ may have an inverse $A^{-1}$ with $A A^{-1} = I_n$ . Only square matrices can be invertible.
Trace: the sum of the diagonal entries, $\text{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}$ .

Watch out

Not every square matrix has an inverse. A square matrix with determinant zero is called singular — it has no inverse. A common exam trap.

Special Square Matrices You Should Know

Type	Definition	Example ( $3 \times 3$ )
Identity $I_n$	1's on diagonal, 0's elsewhere	$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$
Diagonal	non-zero only on diagonal	$\begin{bmatrix}2&0&0\\0&-5&0\\0&0&9\end{bmatrix}$
Symmetric	$a_{ij} = a_{ji}$ for all $i,j$	$\begin{bmatrix}1&4&7\\4&2&5\\7&5&3\end{bmatrix}$

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Square matrix $\iff m=n$ — option (ii).

Concept. A matrix of order $m\times n$ is a square matrix precisely when the number of rows equals the number of columns, i. …

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

Showing the 12 most recent of 13 on this concept.

CBSE 2026Set A1 markMCQ
$A = [a_{ij}]_{m \times n}$
(a) $m = n$
(b) $m < n$
(c) $m > n$
(d) none of these
›Reveal solutionSolution

Square matrix $\Leftrightarrow$ number of rows $=$ number of columns.

An $m\times n$ matrix has $m$ rows and $n$ columns. It is called a square matrix precisely when the number …

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CBSE 2026Set ANNUAL1 markMCQ
$\begin{bmatrix}0 & 0 & 3\\ 0 & 3 & 0\\ 3 & 0 & 0\end{bmatrix}$
(a) Scalar
(b) Diagonal
(c) Unit
(d) Square
›Reveal solutionSolution
The matrix is a plain square matrix — it fails the definitions of diagonal, scalar and unit matrices.

The given matrix is

$M=\begin{bmatrix}0&0&3\\0&3&0\\3&0&0\end{bmatrix}$

A matrix is square if the number of rows equals the number of columns — here $3=3$ , so $M$ is square. ✓

A matrix is diagonal only if every entry off the main diagonal is $0$ . Here the main diagonal entries are $0,3,0$ , but the anti-diagonal entries $M_{13},M_{22},M_{31}=3,3,3$ are non-zero off-diagonal entries (except $M_{22}$ which is on the diagonal) — in particular $M_{13}=3\neq0$ is off-diagonal, so $M$ is not diagonal. …
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CBSE 2026Set ANNUAL1 markMCQ
$A = [a_{ij}]_{m \times n}$
(a) $m < n$
(b) $m > n$
(c) $m = n$
(d) None of these
›Reveal solutionSolution

A square matrix has an equal number of rows and columns.

A matrix $A=[a_{ij}]_{m\times n}$ has $m$ rows and $n$ columns. By definition, it is a square matrix precisely when the number of rows e …

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CBSE 2025Set 65/1/11 markMCQ
$A = \begin{bmatrix} 7 & 0 & x \\ 0 & 7 & 0 \\ 0 & 0 & y \end{bmatrix}$
›Reveal solutionSolution
A scalar matrix is a diagonal matrix with all diagonal entries equal. For the given matrix, this forces $x = 0$ and $y = 7$ , so $y^x = 7^0 = 1$ .

A scalar matrix is a special kind of diagonal matrix — but stricter. In a diagonal matrix, only the entries on the main diagonal can be non-zero; everything else is zero. A scalar matrix goes further: every entry on the main diagonal must be the same constant. That constant is often denoted by $k$ , and the matrix looks like $kI$ , where $I$ is the identity matrix.

So if $A$ is a scalar matrix, all three diagonal entries must be equal. Let’s see what that tells us about $x$ and $y$ .

The matrix is

$A = \begin{bmatrix} 7 & 0 & x \\ 0 & 7 & 0 \\ 0 & 0 & y \end{bmatrix}.$
For $A$ to be a scalar matrix, every off-diagonal entry must be zero — that’s already true except for the $(1,3)$ entry, which is $x$ . So we must have $x = 0$ .

Now the diagonal entries are $7$ , $7$ , and $y$ . For a scalar matrix, all diagonal entries must be equal. So $y$ must equal $7$ .

Therefore $x = 0$ and $y = 7$ . The expression we need is $y^x = 7^0$ . …
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CBSE 2025Set 65/2/11 markMCQ
$A = \operatorname{diag}[3\ \ 5\ \ 2]$
›Reveal solutionSolution
A scalar matrix must have all diagonal entries equal (and off-diagonals zero). Here $A$ has diagonal entries $3,5,2$ — they are not all equal, so $A$ is not a scalar matrix. Assertion (A) is false; Reason (R) is true. The correct option is (D).

The core idea here is the precise definition of a scalar matrix. A diagonal matrix has zeros everywhere except possibly on the main diagonal. A scalar matrix is a special kind of diagonal matrix — one where every diagonal entry is the same number (the scalar). If the diagonal entries are different, it’s just a plain diagonal matrix, not a scalar one.

Let’s check each statement carefully.

Examine Assertion (A): $A = \operatorname{diag}[3\ \ 5\ \ 2]$ means

$A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix}.$
This is certainly a diagonal matrix. But for it to be a scalar matrix, all three diagonal entries must be equal. Here $3 \neq 5 \neq 2$ , so they are not all the same. Therefore $A$ is not a scalar matrix. Assertion (A) is false.

Examine Reason (R):

Reason (R) states: “If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.”

This is exactly the textbook definition. A scalar matrix is $kI_n$ — a diagonal matrix where every diagonal entry is the same non-zero constant $k$ . So Reason (R) is true.

Determine the relationship: …
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CBSE 2025Set ANNUAL1 markMCQ
$A = \begin{bmatrix} 0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0 \end{bmatrix}$
(a) square matrix
(b) diagonal matrix
(c) unit matrix
(d) scalar matrix
›Reveal solutionSolution

Check the definitions: diagonal/scalar/unit matrices all require every off-diagonal entry to be zero, which fails here.

$A = \begin{bmatrix} 0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0 \end{bmatrix}$

A diagonal matrix requires all off-diagonal entries $a_{ij}=0$ for $i\ne j$ . Here $a_{13}=4$ and $a_{31}=4$ are both nonzero, so A is NOT diagonal.

…

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CBSE 2024Set 65/3/11 markMCQ
$Q. Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true. Assertion (A): Every scalar matrix is a diagonal matrix. Reason (R): In a diagonal matrix, all the diagonal elements are 0.$
›Reveal solutionSolution
A scalar matrix is a special diagonal matrix where every diagonal entry is the same constant. The Reason given is false because a diagonal matrix can have any numbers on its diagonal — they are not required to be zero. So Assertion true, Reason false → option (C).

Concept first.

A scalar matrix is a square matrix where every diagonal element is the same scalar $k$ , and all off-diagonal entries are zero. For example,
$\begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix}$
is a scalar matrix.

A diagonal matrix is any square matrix where all entries outside the main diagonal are zero — the diagonal entries can be any numbers (including zero, equal, or all different). So every scalar matrix certainly satisfies the definition of a diagonal matrix (off-diagonals are zero), but the converse is not true.

Now examine the two statements.

Assertion (A): “Every scalar matrix is a diagonal matrix.”

This is true. A scalar matrix has zeros everywhere except on the diagonal, which is exactly the condition for being a diagonal matrix. The fact that the diagonal entries happen to be equal doesn’t break the definition — it only makes it a special case.

Reason (R): “In a diagonal matrix, all the diagonal elements are 0.” …
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CBSE 2024Set D1 markMCQ
$A = [a_{ij}]_{m \times n}$
(a) $m = n$
(b) $m < n$
(c) $m > n$
(d) none of these
›Reveal solutionSolution

Square matrix $\Leftrightarrow m = n$ .

A matrix $A = [a_{ij}]_{m\times n}$ has $m$ rows and $n$ columns. It is called a square matrix when the number of rows equals the number of columns, i.e.

$m = n.$ …

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CBSE 2024Set ANNUAL1 markMCQ
$\begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$
(a) $AB=O, BA=I$
(b) $AB=BA=O$
(c) $AB=BA=I$
(d) $AB=I, BA=O$
(a) diagonal matrix
(b) row matrix
(c) column matrix
(d) unit matrix
›Reveal solutionSolution

A square matrix with all non-diagonal entries zero is a diagonal matrix.

The matrix $\begin{bmatrix}5&0\\0&5\end{bmatrix}$ has both off-diagonal entries equal to $0$ , so it is a diagonal matrix (in fact a scalar matrix). It is not the unit (identity) matrix, since the identity matrix $I_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}$ has $1$ 's on the diagonal, not $5$ 's.

…

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CBSE 2023Set ANNUAL1 mark
$Q. Write the number of entries in a square matrix of order 4.$
›Reveal solutionSolution

A square matrix of order $n$ has $n^2$ entries, so order 4 gives 16 entries.

A square matrix of order $n$ has $n$ rows and $n$ columns, giving $n \times n = n^2$ entries in total. For $n = 4$ , …

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CBSE 2021Set ANNUAL1 markMCQ
$A=[a_{ij}]_{m\times n}$
(a) $m<n$
(b) $m>n$
(c) $m=n$
(d) none of these
›Reveal solutionSolution

A matrix $A=[a_{ij}]_{m\times n}$ is a square matrix when the number of rows equals the number of columns.

…

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CBSE 2019Set ANNUAL1 mark
$Q. Define a scalar matrix.$
›Reveal solutionSolution

A scalar matrix is a diagonal matrix whose diagonal entries are all equal to one constant $k$ .

Concept. A diagonal matrix has all non-diagonal entries zero. A scalar matrix is a diagonal matrix in which, in addition, every diagonal entry is the same number.

Definition. A square matrix $A=[a_{ij}]$ is a scalar matrix if

$a_{ij}=\begin{cases}0, & i e j\\[2pt] k, & i=j\end{cases}$

for some fixed scalar $k$ . Equivalently $A=kI$ .

…

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