UG Mathematics Booster Test 2 - Types of Matrices -

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UG Mathematics Booster Test 2 - Types of Matrices

⏱️ 30 Minutes

❓ 20 Questions

📌 Answers are locked once submitted — results and explanations appear at the end.

UG Mathematics Booster Test 2 - Types of Matrices

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QUESTION 1 OF 20

A force vector in physics is represented as a column matrix
\(\left[\begin{aligned}F_{x}\\ F_{y}\\ F_{z}\end{aligned}\right]\)
Which category defines this matrix?

Answer Review

1 A force vector in physics is represented as a column matrix
\(\left[\begin{aligned}F_{x}\\ F_{y}\\ F_{z}\end{aligned}\right]\)
Which category defines this matrix?

A Row matrix

B Square matrix

C Diagonal matrix

D Column matrix

💡 Short Explanation

Matrix has one vertical column. Elements are arranged downward. Hence it is a column matrix.

✅ Correct Answer Explanation

A column matrix contains exactly one column and any number of rows. The given force vector has three rows and one column: \(3\times 1\) Hence it is a column matrix. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Row matrices contain one horizontal row.
Option B → Square matrices require equal rows and columns.
Option C → Diagonal matrices must be square matrices.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Observe the vertical arrangement of elements.

Final Logic:

Single-column arrangement defines a column matrix.

🧠 Memory Trick

"Vertical → Column"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Column Matrix

2 A factory stores quantities of 5 mixtures in an \(m\times 1\) matrix. If total elements = 5, find \(m\).

A 1

B 2

C 5

D 10

💡 Short Explanation

Matrix order is \(m\times 1\). Total elements = \(m\times 1\). Therefore \(m=5\).

✅ Correct Answer Explanation

For a matrix of order: \(m\times 1\) the total number of elements equals: \(m⋅1=m\) Given total elements = 5, therefore: \(m=5\) Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Gives only one element.
Option B → Produces two elements only.
Option D → Produces ten elements, not five.

🎯 Strategy

Used: Substitution

Application:

Use total elements = rows × columns.

Final Logic:

\(m\times 1=5\Rightarrow m=5\)

🧠 Memory Trick

"Column matrix → rows decide total"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Order of Matrix

3 Identify the incorrect statement about a row matrix:

A It has exactly one row

B It can have zero columns

C It is of the form \(1\times n\)

D Elements are arranged horizontally

💡 Short Explanation

Row matrices need one row and n columns. Matrices cannot have zero columns. Horizontal arrangement defines row matrices.

✅ Correct Answer Explanation

A row matrix is defined as a matrix having one row and n columns: \(1\times n\) Matrices must contain at least one column. Hence the statement "It can have zero columns" is incorrect. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Correct definition of a row matrix.
Option C → Standard order representation for row matrices.
Option D → Row matrix entries are arranged horizontally.

🎯 Strategy

Used: Elimination

Application:

Check which statement violates matrix definition.

Final Logic:

Matrices cannot exist with zero columns.

🧠 Memory Trick

"Row → one horizontal line"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Row Matrix

4 A row matrix of order \(1\times 2\) has entries chosen from \(\left\{0,\ 1\right\}\).
Probability that it is a zero matrix?
Total possibilities = \(2^{2}=4\)
Only one is zero matrix → probability = \(1/4\)

A \(1/4\)

B \(1/2\)

C \(3/4\)

D 1

💡 Short Explanation

Two entries each have two choices. Total matrices = 4. Only one has both zeros.

✅ Correct Answer Explanation

Possible matrices are: \(\left[0 0],[0 1],[1 0],[1 1\right]\) Only: \(\left[0 0\right]\) is a zero matrix. Thus probability equals: \(\frac{1}{4}\) Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Assumes two favorable outcomes incorrectly.
Option C → Counts nonzero matrices instead.
Option D → Probability cannot equal certainty here.

🎯 Strategy

Used: Substitution

Application:

Count favorable outcomes over total outcomes.

Final Logic:

1 favorable case out of 4 total cases.

🧠 Memory Trick

"Only all-zero matrix counts"

📚 Class XI – NCERT Mathematics – Chapter: Probability – Topic: Classical Probability

5 For a square matrix, which operations are always defined?
I. Multiplication with itself
II. Addition with identity matrix
III. Transpose gives same order

A I only

B II and III only

C I and III only

D I, II, and III

💡 Short Explanation

Square matrices allow self multiplication. Identity matrix has same order. Transpose preserves dimensions.

✅ Correct Answer Explanation

A square matrix has equal rows and columns: \(n\times n\) Therefore: Self multiplication is defined. Addition with identity matrix of same order is defined. Transpose remains \(n\times n\). Hence all statements are true. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores addition and transpose properties.
Option B → Multiplication with itself is also defined.
Option C → Addition with identity matrix is valid.

🎯 Strategy

Used: Option Grouping

Application:

Check compatibility conditions for operations.

Final Logic:

All operations preserve square order.

🧠 Memory Trick

"Square matrices work with themselves"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Operations on Matrices

6 For a diagonal matrix
\(\left[\begin{pmatrix}a & 0\\ 0 & b\end{pmatrix}\right]\)
the determinant \(=ab\) represents area. If \(a=3,b=4\), find area.

A 7

B 10

C 12

D 24

💡 Short Explanation

Determinant equals product of diagonal entries. Multiply 3 and 4. Area obtained is 12.

✅ Correct Answer Explanation

For a diagonal matrix: \(det⁡(A)=ab\) Substituting: \(3\times 4=12\) Hence the determinant, representing area scaling, equals 12. Therefore Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Gives sum instead of product.
Option B → Incorrect multiplication.
Option D → Doubles the correct value.

🎯 Strategy

Used: Substitution

Application:

Insert values directly into determinant formula.

Final Logic:

Determinant of diagonal matrix = product of diagonal entries.

🧠 Memory Trick

"Diagonal determinant → multiply diagonals"

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Determinant of Diagonal Matrix

7 In a diagonal matrix \(A\), all non-diagonal elements \(a_{ij}\)(for \(i\neq j\)) must be zero.
If a non-diagonal element is given by an expression involving a variable, what must its value be for \(A\) to remain diagonal?

A 1

B 0

C Positive value

D Any real number

💡 Short Explanation

Non-diagonal entries in diagonal matrices are zero. Any variable expression must evaluate to zero. Otherwise matrix ceases to be diagonal.

✅ Correct Answer Explanation

A diagonal matrix satisfies: \(a_{ij}=0for i\neq j\) Thus every non-diagonal entry must be zero, regardless of how it is expressed algebraically. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Non-diagonal entries cannot equal 1 generally.
Option C → Positive values violate diagonal form.
Option D → Arbitrary real numbers are not allowed.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Use the definition of diagonal matrices directly.

Final Logic:

Diagonal matrices require off-diagonal zeros.

🧠 Memory Trick

"Off-diagonal → always zero"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Diagonal Matrix

8 Arrange the matrices in ascending order of the number of main diagonal elements:
I. Diagonal matrix
II. Scalar matrix
III. Identity matrix

A II, I, III

B I, II, III

C III, I, II

D II, III, I

💡 Short Explanation

Every square matrix of same order has same diagonal count. Diagonal, scalar, and identity matrices are square matrices. Hence no ascending distinction exists.

✅ Correct Answer Explanation

Diagonal, scalar, and identity matrices are all square matrices. A square matrix of order n always contains: \(n\) main diagonal elements. Hence all three types have equal counts of diagonal entries. Therefore no arrangement is mathematically meaningful.

❓ Why Other Options Are Incorrect

Option A → Assumes different diagonal counts incorrectly.
Option B → No matrix here has fewer diagonal positions.
Option C → Identity and scalar matrices still have same diagonal count.

🎯 Strategy

Used: Elimination

Application:

Compare structural definitions instead of names.

Final Logic:

All square matrices of same order have equal diagonal positions.

🧠 Memory Trick

"Same order → same diagonal count"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Special Matrices

9 A scalar matrix scales a point \(\left(x,\ y\right)\)to \(\left(2x,\ 2y\right)\). What is the diagonal value \(k\)?

A 1

B 3

C 2

D 4

💡 Short Explanation

Scalar matrices multiply coordinates uniformly. Both coordinates doubled. Therefore scaling factor equals 2.

✅ Correct Answer Explanation

A scalar matrix has the form: \(kI\) Multiplying a point by this matrix scales each coordinate by k. Since: \(\left(x,y)\rightarrow (2x,2y\right)\) we get: \(k=2\) Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Leaves coordinates unchanged.
Option B → Would produce \(\left(3x,\ 3y\right)\).
Option D → Would quadruple scaling effect.

🎯 Strategy

Used: Substitution

Application:

Compare transformed coordinates with original coordinates.

Final Logic:

Scaling factor equals coordinate multiplier.

🧠 Memory Trick

"Scalar matrix = same scaling everywhere"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Scalar Matrix

10 Assertion (A): An identity matrix is not a scalar matrix.
Reason (R): A scalar matrix has diagonal entries equal to a constant \(k\), and for identity matrix \(k=1\).

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Identity matrix is a scalar matrix. Here scalar value is \(k=1\). Reason is true but Assertion is false.

✅ Correct Answer Explanation

A scalar matrix has equal diagonal entries and zero elsewhere: \(kI\) For the identity matrix: \(k=1\) Hence every identity matrix is a scalar matrix. Therefore Assertion is false and Reason is true. Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Reason statement is mathematically true.
Option B → Identity matrices are scalar matrices.
Option C → Assertion itself is incorrect.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Relate identity matrix to scalar matrix definition.

Final Logic:

Identity matrix is a special scalar matrix with \(k=1\).

🧠 Memory Trick

"Identity = scalar with k = 1"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Scalar and Identity Matrix

11 Match the Following

List I	List II
1. \(a_{ii}\)	a. 1
2. \(a_{ij}\)for \(i\neq j\)	b. 0
3. Diagonal element	c. 1
4. Non-diagonal element	d. 0

A 1-a, 2-b, 3-c, 4-d

B 1-b, 2-a, 3-d, 4-c

C 1-d, 2-c, 3-b, 4-a

D 1-c, 2-d, 3-a, 4-b

💡 Short Explanation

Identity matrix has diagonal ones. Non-diagonal entries are zero. Match diagonal with 1 and others with 0.

✅ Correct Answer Explanation

For an identity matrix: \(a_{ii}=1,a_{ij}=0 for i\neq j\) Thus diagonal elements correspond to 1 and non-diagonal elements correspond to 0. Therefore the correct matching is: 1-a, 2-b, 3-c, 4-d Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Reverses diagonal and non-diagonal values.
Option C → Incorrectly matches zero with diagonal entries.
Option D → Non-diagonal entries cannot be 1 in identity matrices.

🎯 Strategy

Used: Option Grouping

Application:

Match identity matrix properties systematically.

Final Logic:

Diagonal → 1, non-diagonal → 0.

🧠 Memory Trick

"Identity = ones on diagonal"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Identity Matrix

12 In an identity matrix of order \(n\), the number of non-diagonal (zero) elements is:

A \(n\)

B \(n^{2}-n\)

C \(n^{2}\)

D \(n+1\)

💡 Short Explanation

Total elements are \(n^{2}\). Diagonal elements are n. Remaining entries are non-diagonal zeros.

✅ Correct Answer Explanation

An identity matrix of order \(n\) contains: \(n^{2}\) total elements. Out of these, n are diagonal ones. Therefore the number of non-diagonal zero entries is: \(n^{2}-n\) Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Counts only diagonal entries.
Option C → Represents total elements, not non-diagonal entries.
Option D → No matrix property gives \(n+1\) here.

🎯 Strategy

Used: Substitution

Application:

Subtract diagonal entries from total entries.

Final Logic:

Non-diagonal entries = total − diagonal.

🧠 Memory Trick

"Total minus diagonal"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Identity Matrix

📄 Passage

A zero (null) matrix has all elements equal to zero and acts as the additive identity.

If a zero matrix is added to a matrix A, the result is:

A Zero matrix

B Identity matrix

C Matrix A

D Undefined

💡 Short Explanation

Zero matrix acts as additive identity. Adding zero changes nothing. Result remains matrix A.

✅ Correct Answer Explanation

The zero matrix behaves like 0 in ordinary arithmetic. Therefore: \(A+O=A\) where O denotes the zero matrix. Hence adding a zero matrix to A leaves A unchanged. Therefore Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Result becomes zero only if A itself is zero.
Option B → Identity matrices relate to multiplication, not addition.
Option D → Matrix addition is well-defined for same order matrices.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Recall additive identity property of zero matrices.

Final Logic:

Adding zero leaves the matrix unchanged.

🧠 Memory Trick

"Matrix + zero = same matrix"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Zero Matrix

📄 Passage

A zero (null) matrix has all elements equal to zero and acts as the additive identity.

If the product of two matrices is a zero matrix, what must be true?

A First matrix is zero

B Second matrix is zero

C Both are zero

D Neither needs to be zero

💡 Short Explanation

Product may be zero without zero matrices. Nonzero matrices can multiply to zero. Such matrices are called zero divisors.

✅ Correct Answer Explanation

In matrix algebra, it is possible for two nonzero matrices to produce a zero matrix when multiplied. Therefore neither matrix must necessarily be zero. Hence Option D is correct. Example: \(AB=OwithA\neq O, B\neq O\)

❓ Why Other Options Are Incorrect

Option A → First matrix need not be zero.
Option B → Second matrix also need not be zero.
Option C → Both matrices can be nonzero.

🎯 Strategy

Used: Odd One Out

Application:

Use matrix multiplication properties beyond ordinary arithmetic.

Final Logic:

Zero product does not force zero matrices.

🧠 Memory Trick

"Zero product rule fails for matrices"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Matrix Multiplication

15 If diagonal elements are 2, 3, 5, what is their sum?

A 10

B 9

C 8

D 7

💡 Short Explanation

Add all diagonal entries directly. \(2+3+5=10\). Sum of diagonal elements equals trace.

✅ Correct Answer Explanation

The required sum is: \(2+3+5=10\) Therefore the sum of diagonal elements equals 10. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Misses one diagonal entry.
Option C → Arithmetic addition is incorrect.
Option D → Too small compared to actual total.

🎯 Strategy

Used: Substitution

Application:

Add the listed diagonal entries carefully.

Final Logic:

Trace equals the sum of principal diagonal entries.

🧠 Memory Trick

"Trace = diagonal total"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Principal Diagonal

16 The elements of the principal diagonal of a square matrix are given as \(a_{11},a_{22},a_{33},a_{44}\). What is the minimum possible order of the matrix?

A Order 2

B Order 4

C Order 3

D Order 1

💡 Short Explanation

Largest diagonal index is 4. Matrix must contain 4 rows and columns. Minimum order is 4×4.

✅ Correct Answer Explanation

Since the principal diagonal includes \(a_{44}\), the matrix must contain at least 4 rows and 4 columns. Therefore the minimum order is: \(4\times 4\) Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Order 2 lacks \(a_{33}\)and \(a_{44}\).
Option C → Order 3 lacks the element \(a_{44}\).
Option D → Order 1 contains only \(a_{11}\).

🎯 Strategy

Used: Elimination

Application:

Check the highest row-column index present.

Final Logic:

Largest index determines minimum order.

🧠 Memory Trick

"Highest subscript decides size"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Principal Diagonal

17 Which properties apply to a square matrix?
I. It has equal number of rows and columns
II. It can be a diagonal matrix
III. It can be an identity matrix

A I only

B I and II only

C II and III only

D I, II, and III

💡 Short Explanation

Square matrices have equal dimensions. Diagonal matrices are square matrices. Identity matrices are also square matrices.

✅ Correct Answer Explanation

A square matrix satisfies: \(m=n\) It may also be diagonal or identity because both are special forms of square matrices. Therefore all three statements are true. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores diagonal and identity possibilities.
Option B → Identity matrices are also square matrices.
Option C → Equal dimensions are the defining property.

🎯 Strategy

Used: Option Grouping

Application:

Verify all valid properties of square matrices.

Final Logic:

Diagonal and identity matrices are subclasses of square matrices.

🧠 Memory Trick

"Identity and diagonal are square"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Square Matrix

18 If a matrix has equal diagonal elements and zero elsewhere, what type is it?

A Null matrix

B Column matrix

C Scalar matrix

D Identity matrix

💡 Short Explanation

Scalar matrices have equal diagonal entries. All non-diagonal entries are zero. Identity matrix is a special scalar matrix.

✅ Correct Answer Explanation

A scalar matrix is defined as a diagonal matrix whose diagonal entries are equal: \(\left[\begin{pmatrix}k & 0\\ 0 & k\end{pmatrix}\right]\) Therefore Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Null matrices require all entries zero.
Option B → Column matrices have one column only.
Option D → Identity matrices specifically require diagonal entries equal to 1.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Match the matrix description with standard matrix forms.

Final Logic:

Equal diagonal entries define scalar matrices.

🧠 Memory Trick

"Scalar → same diagonal constant"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Scalar Matrix

19 Which statement is incorrect?

A Every identity matrix is a scalar matrix

B Every scalar matrix is an identity matrix

C Every scalar matrix is a diagonal matrix

D Every diagonal matrix is a square matrix

💡 Short Explanation

Identity matrices are scalar matrices with \(k=1\). Scalar matrices may have any constant k. Hence not every scalar matrix is identity.

✅ Correct Answer Explanation

Every identity matrix is a scalar matrix because its diagonal entries are all 1. However scalar matrices may have any constant diagonal value \(k\). Therefore not every scalar matrix is an identity matrix. Hence Option B is the incorrect statement.

❓ Why Other Options Are Incorrect

Option A → Identity matrices are scalar matrices with \(k=1\).
Option C → Scalar matrices are always diagonal matrices.
Option D → Diagonal matrices must necessarily be square.

🎯 Strategy

Used: Odd One Out

Application:

Compare hierarchy among matrix types.

Final Logic:

Identity matrix is a special scalar matrix, not vice versa.

🧠 Memory Trick

"Identity ⊂ Scalar"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Special Matrices

20 Arrange from most general to most specific:
I. Square Matrix
II. Scalar Matrix
III. Identity Matrix
IV. Diagonal Matrix

A I, IV, II, III

B I, II, III, IV

C IV, I, II, III

D III, II, IV, I

💡 Short Explanation

Square matrices are most general. Diagonal matrices are special square matrices. Identity matrices are most specific.

✅ Correct Answer Explanation

The hierarchy is: \(Square Matrix\supset Diagonal Matrix\supset Scalar Matrix\supset Identity Matrix\) Thus the correct arrangement from most general to most specific is: I, IV, II, III Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Places scalar matrices before diagonal matrices incorrectly.
Option C → Diagonal matrices are not more general than square matrices.
Option D → Reverses the hierarchy completely.

🎯 Strategy

Used: Option Grouping

Application:

Arrange matrix types according to subset relationships.

Final Logic:

Identity matrices are the most specialized form.

🧠 Memory Trick

"Square → Diagonal → Scalar → Identity"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Classification of Matrices

UG Mathematics Booster Test 2 - Types of Matrices

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