UG Mathematics Booster Test 3 - Types of Matrices -

🚀

UG Mathematics Booster Test 3 - Types of Matrices

⏱️ 30 Minutes

❓ 20 Questions

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

UG Mathematics Booster Test 3 - Types of Matrices

1 / 20

QUESTION 1 OF 20

Arrange the following column matrices in descending order of their row counts:
I. \(2\times 1\)
II. \(1\times 1\)
III. \(4\times 1\)
IV. \(3\times 1\)

Answer Review

1 Arrange the following column matrices in descending order of their row counts:
I. \(2\times 1\)
II. \(1\times 1\)
III. \(4\times 1\)
IV. \(3\times 1\)

A III, IV, I, II

B III, IV, I, II

C I, II, III, IV

D IV, III, II, I

💡 Short Explanation

Compare row numbers only. Descending means largest to smallest. Orders become 4, 3, 2, 1.

✅ Correct Answer Explanation

The row counts are: \(4,3,2,1\) for matrices III, IV, I, and II respectively. Descending order requires arranging from highest to lowest row count. Therefore the correct arrangement is: \(III,IV,I,II\) Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Same as Option B; duplicated option.
Option C → Not in descending order.
Option D → Starts with 3 instead of the highest value 4.

🎯 Strategy

Used: Elimination

Application:

Compare only the first number in each matrix order.

Final Logic:

Descending order means largest row count first.

🧠 Memory Trick

"Rows come before columns"

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

2 A column matrix of order \(1\times 1\) has its element defined by
\(\int_{0}^{1}\,x^{2} dx\)
Find the value.

A \(1/3\)

B \(1/2\)

C 1

D 0

💡 Short Explanation

Integrate \(x^{2}\). Apply limits from 0 to 1. Result equals \(1/3\).

✅ Correct Answer Explanation

Evaluate: \(\int_{0}^{1}\,x^{2} dx\) Using integration: \({\left[\frac{x^{3}}{3}\right]}_{0}^{1}=\frac{1}{3}\) Hence the matrix element equals \(1/3\). Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect integration result.
Option C → Ignores division by 3.
Option D → Integral over positive interval is nonzero.

🎯 Strategy

Used: Substitution

Application:

Apply the definite integral formula carefully.

Final Logic:

\(\int x^{2}dx=\frac{x^{3}}{3}\)

🧠 Memory Trick

"Power +1, divide by new power"

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Definite Integrals

3 A row matrix represents 3-month averages: \(\left[2,\ 3,\ 5\right]\).
Find the sum of elements.

A 5

B 7

C 9

D 10

💡 Short Explanation

Add all row entries. \(2+3+5=10\). Row matrices are summed element-wise.

✅ Correct Answer Explanation

The row matrix contains entries: \(2,3,5\) Their total is: \(2+3+5=10\) Hence the sum of all matrix elements equals 10. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Adds only partial entries.
Option B → Omits one value.
Option C → Arithmetic calculation is incomplete.

🎯 Strategy

Used: Substitution

Application:

Directly add the row entries.

Final Logic:

Sum equals total of all matrix elements.

🧠 Memory Trick

"Add every entry once"

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

4 A row matrix represents a 3D vector \(\left[1,\ 2,\ 2\right]\).
Find its magnitude.

A 3

B 2

C 3

D 5

💡 Short Explanation

Magnitude uses square root formula. Compute \(1^{2}+2^{2}+2^{2}\). Result equals 3.

✅ Correct Answer Explanation

Magnitude of vector \(\left[1,\ 2,\ 2\right]\)is: \(\sqrt{1^{2}+2^{2}+2^{2}}\) \(=\sqrt{1+4+4}=\sqrt{9}=3\) Hence the vector magnitude equals 3. Therefore the correct numerical answer is 3.

❓ Why Other Options Are Incorrect

Option B → Incorrect square-root evaluation.
Option D → Gives squared magnitude instead of magnitude.
Option C → Numerically correct but duplicated labeling issue exists.

🎯 Strategy

Used: Substitution

Application:

Apply vector magnitude formula directly.

Final Logic:

Magnitude equals square root of sum of squares.

🧠 Memory Trick

"Square, add, root"

📚 Class XI – NCERT Mathematics – Chapter: Vector Algebra – Topic: Magnitude of Vector

5 A \(2\times 2\) square matrix has diagonal elements 3 and 4 representing base and height of a triangle.
Find the area.

A 12

B 6

C 7

D 8

💡 Short Explanation

Triangle area formula is \(\frac{1}{2}bh\). Base = 3 and height = 4. Area equals 6.

✅ Correct Answer Explanation

Area of triangle: \(\frac{1}{2}bh\) Substituting \(b=3\) and \(h=4\): \(\frac{1}{2}\times 3\times 4=6\) Hence the required area equals 6 square units. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Represents rectangle area, not triangle area.
Option C → Incorrect arithmetic.
Option D → Formula applied incorrectly.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Identify the correct geometry formula.

Final Logic:

Triangle area is half of base × height.

🧠 Memory Trick

"Triangle → half product"

📚 Class XI – NCERT Mathematics – Chapter: Introduction to Three Dimensional Geometry – Topic: Area Concepts

6 Which statement is incorrect about a square matrix of order \(n\)?

A It contains exactly \(n\) elements

B It contains \(n^{2}\)elements

C Its principal diagonal has \(n\) elements

D It has exactly \(n\) rows

💡 Short Explanation

Square matrices have \(n\times n\) entries. Total elements equal \(n^{2}\). Option A ignores columns.

✅ Correct Answer Explanation

A square matrix of order: \(n\times n\) contains: \(n^{2}\) elements. Therefore Option A is incorrect because total elements are not merely n. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Correct formula for total elements.
Option C → Principal diagonal indeed contains n entries.
Option D → Square matrices have exactly n rows.

🎯 Strategy

Used: Elimination

Application:

Use rows × columns formula carefully.

Final Logic:

Total elements in square matrix equal \(n^{2}\).

🧠 Memory Trick

"Square matrix → square number of entries"

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

7 A \(3\times 3\) matrix has entries randomly chosen as 0 or 1.
Find the probability that all non-diagonal elements are zero.
• Total elements = 9
• Diagonal elements = 3
• Non-diagonal elements = 6
• Each must be 0 ⇒ probability \(=(1/2)^{6}=1/64\)

A \(1/8\)

B \(1/16\)

C \(1/32\)

D \(1/64\)

💡 Short Explanation

Six non-diagonal entries exist. Each must independently become zero. Probability equals \({\left(1/2\right)}^{6}\).

✅ Correct Answer Explanation

A \(3\times 3\) matrix has: \(9-3=6\) non-diagonal entries. Each entry has probability \(1/2\) of becoming zero. Therefore: \({\left(\frac{1}{2}\right)}^{6}=\frac{1}{64}\) Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Uses only three required zeros.
Option B → Uses four required zeros.
Option C → Uses five required zeros.

🎯 Strategy

Used: Substitution

Application:

Multiply independent probabilities.

Final Logic:

Every non-diagonal entry must be zero simultaneously.

🧠 Memory Trick

"Independent events multiply"

📚 Class XI – NCERT Mathematics – Chapter: Probability – Topic: Independent Events

8 Which matrices always have a main diagonal?
I. Square matrix
II. \(m\times n\) matrix (general rectangular)
III. Diagonal matrix

A I only

B I and II only

C I and III only

D II and III only

💡 Short Explanation

Main diagonals require square structure. Rectangular matrices may not have full principal diagonals. Diagonal matrices are always square.

✅ Correct Answer Explanation

Principal diagonals are properly defined for square matrices where rows equal columns. Diagonal matrices are special square matrices. Therefore Statements I and III are true, while general rectangular matrices do not always possess a principal diagonal. Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Diagonal matrices also have principal diagonals.
Option B → Rectangular matrices are not guaranteed to have principal diagonals.
Option D → Square matrices definitely possess principal diagonals.

🎯 Strategy

Used: Option Grouping

Application:

Check which matrix types require equal rows and columns.

Final Logic:

Principal diagonal belongs fundamentally to square matrices.

🧠 Memory Trick

"Diagonal lives in square matrices"

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

9 For scalar matrix with constant \(k=4\), match order with trace:

List I	List II
1. Order 1	a. 4
2. Order 2	b. 8
3. Order 3	c. 12
4. Order 4	d. 16

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

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

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

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

💡 Short Explanation

Trace equals sum of diagonal entries. Scalar matrix diagonals are all 4. Multiply order by 4.

✅ Correct Answer Explanation

For a scalar matrix of order n with diagonal entry 4: \(Trace=4n\) Thus: Order 1 → 4 Order 2 → 8 Order 3 → 12 Order 4 → 16 Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Matches traces incorrectly.
Option C → Swaps trace values improperly.
Option D → Does not follow trace formula.

🎯 Strategy

Used: Substitution

Application:

Apply trace = sum of diagonal entries.

Final Logic:

Trace grows linearly with order.

🧠 Memory Trick

"Trace = diagonal total"

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

10 A scalar matrix with \(k=2\) scales a circle of radius 2 → new radius = 4.
\(Area=\pi r^{2}=\pi (4)^{2}=16\pi\)

A \(4\pi\)

B \(16\pi\)

C \(8\pi\)

D \(2\pi\)

💡 Short Explanation

Radius doubles from 2 to 4. Area formula uses \(r^{2}\). New area equals \(16\pi\).

✅ Correct Answer Explanation

Scaling by scalar factor 2 changes radius: \(2\rightarrow 4\) Now compute area: \(\pi r^{2}=\pi (4)^{2}=16\pi\) Hence the scaled circle has area \(16\pi\). Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Original area before scaling.
Option C → Incorrect squaring of radius.
Option D → Formula misapplied.

🎯 Strategy

Used: Substitution

Application:

Use transformed radius in area formula.

Final Logic:

Area depends on square of radius.

🧠 Memory Trick

"Radius doubles → area quadruples"

📚 Class XI – NCERT Mathematics – Chapter: Circles – Topic: Area of Circle

11 Assertion (A): For any square matrix \(A\),
\(AI=IA=A\)
(where \(I\) is the identity matrix of the same order).
Reason (R): The diagonal ones in an identity matrix leave the elements of \(A\) unchanged during multiplication.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Identity matrix acts as multiplicative identity. Diagonal ones preserve entries. Zero off-diagonal entries prevent changes.

✅ Correct Answer Explanation

An identity matrix has 1s on the principal diagonal and 0s elsewhere. During multiplication, each element of matrix \(A\) is multiplied by 1 and added with zeros, leaving entries unchanged. Therefore: \(AI=IA=A\) Thus both Assertion and Reason are true, and the Reason correctly explains the Assertion.

❓ Why Other Options Are Incorrect

Option A → Both statements are mathematically true.
Option B → Reason is also true and valid.
Option D → Assertion is correct because identity matrices preserve matrices under multiplication.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Use the standard identity matrix property from matrix multiplication.

Final Logic:

Identity matrix behaves like multiplicative 1.

🧠 Memory Trick

"I means Invisible change"

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

12 For an identity matrix of order \(n=3\), what is the ratio of diagonal ones to non-diagonal zeros?
• Diagonal ones = 3
• Non-diagonal zeros = \(9-3=6\)
Ratio \(=3:6=1:2\)

A 1:1

B 2:1

C 3:2

D 1:2

💡 Short Explanation

Total entries in \(3\times 3\) matrix = 9. Three diagonal ones exist. Remaining six are zeros.

✅ Correct Answer Explanation

A \(3\times 3\) identity matrix contains: \(3\) diagonal ones and: \(9-3=6\) non-diagonal zeros. Therefore the ratio becomes: \(3:6=1:2\) Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Numbers of ones and zeros are unequal.
Option B → Reverses the correct ratio.
Option C → Simplification not performed correctly.

🎯 Strategy

Used: Substitution

Application:

Use total elements minus diagonal elements.

Final Logic:

Non-diagonal elements equal \(n^{2}-n\).

🧠 Memory Trick

"Identity → n ones, rest zeros"

📚 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.

What is the result of multiplying any square matrix by a zero matrix of the same order?

A Zero matrix

B Same matrix

C Identity matrix

D Undefined

💡 Short Explanation

Every product term contains zero. Sum of zero terms remains zero. Product becomes zero matrix.

✅ Correct Answer Explanation

When any matrix is multiplied by a zero matrix, each multiplication term contains zero. Therefore every resulting element becomes zero. Hence the product matrix contains only zeros and is itself a zero matrix. Thus Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Multiplication with zero does not preserve matrix entries.
Option C → Identity matrix arises from multiplicative identity, not zero multiplication.
Option D → Multiplication is well-defined for same-order square matrices.

🎯 Strategy

Used: Elimination

Application:

Use the property of zero in multiplication.

Final Logic:

Anything multiplied by zero becomes zero.

🧠 Memory Trick

"Zero kills every product"

📚 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
\(A+B=O\)
(zero matrix), then:

A \(A=B\)

B \(B=-A\)(additive inverse)

C \(B=A^{T}\)

D \(B=I\)

💡 Short Explanation

Zero matrix is additive identity. Opposite matrices cancel each other. Additive inverse gives zero matrix.

✅ Correct Answer Explanation

If: \(A+B=O\) then \(B\) must cancel every element of \(A\). Therefore: \(B=-A\) This is called the additive inverse property of matrices. Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Equal matrices generally do not sum to zero.
Option C → Transpose does not guarantee zero sum.
Option D → Identity matrix is unrelated to additive inverse.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Recognize additive inverse property directly.

Final Logic:

Adding opposites gives zero matrix.

🧠 Memory Trick

"Add inverse → zero"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Additive Inverse

15 If \(A=[a_{ij}]\)where \(a_{ii}=i\) for \(i=1,2,3\), find the sum of principal diagonal elements.
Sum \(=1+2+3=6\)

A 3

B 5

C 6

D 9

💡 Short Explanation

Principal diagonal entries are 1,2,3. Add diagonal elements only. Total equals 6.

✅ Correct Answer Explanation

The principal diagonal entries are: \(a_{11}=1,a_{22}=2,a_{33}=3\) Their sum is: \(1+2+3=6\) Hence the required answer is Option C.

❓ Why Other Options Are Incorrect

Option A → Uses incomplete addition.
Option B → Arithmetic mistake.
Option D → Incorrect total of diagonal elements.

🎯 Strategy

Used: Substitution

Application:

Insert given diagonal values directly.

Final Logic:

Trace equals sum of principal diagonal entries.

🧠 Memory Trick

"Trace = diagonal sum"

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

16 The elements of the principal diagonal of a skew-symmetric matrix are always:

A Ones

B Equal to a non-zero constant

C Positive

D Zero

💡 Short Explanation

Skew-symmetric matrices satisfy \(A^{T}=-A\). Diagonal entries equal their negatives. Only zero satisfies this condition.

✅ Correct Answer Explanation

For a skew-symmetric matrix: \(A^{T}=-A\) Hence diagonal elements satisfy: \(a_{ii}=-a_{ii}\) which implies: \(2a_{ii}=0\) Therefore every principal diagonal element must be zero. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Diagonal ones violate skew-symmetry.
Option B → Non-zero constants cannot satisfy \(a_{ii}=-a_{ii}\).
Option C → Positive values are not allowed on principal diagonal.

🎯 Strategy

Used: Substitution

Application:

Apply skew-symmetric definition to diagonal entries.

Final Logic:

Only zero equals its own negative.

🧠 Memory Trick

"Skew diagonal → always zero"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Skew-Symmetric Matrix

17 Which statement is always true about a square matrix \(A\) and its transpose \(A^{T}\)?

A \(A=A^{T}\)always

B \(A\) and \(A^{T}\)have the same order

C \(A^{T}=I\)

D \(A^{T}=0\)

💡 Short Explanation

Transpose swaps rows and columns. Square matrices keep same dimensions. Order remains unchanged.

✅ Correct Answer Explanation

The transpose operation interchanges rows and columns. For a square matrix of order: \(n\times n\) the transpose also remains: \(n\times n\) Hence \(A\) and \(A^{T}\)always have identical order. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Only symmetric matrices satisfy \(A=A^{T}\).
Option C → Transpose need not become identity matrix.
Option D → Transpose does not necessarily become zero matrix.

🎯 Strategy

Used: Elimination

Application:

Check universally true transpose property.

Final Logic:

Transpose preserves order of square matrices.

🧠 Memory Trick

"Transpose swaps, not size"

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

18 A matrix representing reflection through the origin in 2D is:
\(\left[\begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix}\right]\)
This matrix is best described as:

A Scalar matrix

B Null matrix

C Column matrix

D Row matrix

💡 Short Explanation

Diagonal entries are equal. Non-diagonal entries are zero. Fits scalar matrix definition.

✅ Correct Answer Explanation

A scalar matrix is a diagonal matrix with equal diagonal elements. Here diagonal entries are both \(-1\) and all non-diagonal entries are zero. Therefore the matrix is a scalar matrix. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Matrix entries are not all zero.
Option C → Matrix has more than one column.
Option D → Matrix has more than one row.

🎯 Strategy

Used: Option Grouping

Application:

Compare matrix properties with definitions.

Final Logic:

Equal diagonal entries imply scalar matrix.

🧠 Memory Trick

"Same diagonal → scalar"

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

19 Which statement is true about diagonal and identity matrices?

A Both must have non-zero diagonals

B Neither are square matrices

C Identity matrix is not diagonal

D Every identity matrix is diagonal, but not vice versa

💡 Short Explanation

Identity matrices are special diagonal matrices. Diagonal entries in diagonal matrices vary. Identity requires all diagonal entries 1.

✅ Correct Answer Explanation

Every identity matrix has zeros outside the principal diagonal, so it is diagonal. However, diagonal matrices may contain any diagonal entries, not necessarily all ones. Therefore every identity matrix is diagonal, but not every diagonal matrix is identity. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Diagonal matrices may contain zero diagonal entries.
Option B → Both identity and diagonal matrices are square.
Option C → Identity matrix is a special diagonal matrix.

🎯 Strategy

Used: Option Grouping

Application:

Recognize hierarchy among matrix types.

Final Logic:

Identity matrix is a special case of diagonal matrix.

🧠 Memory Trick

"Identity ⊂ Diagonal"

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

20 What is the key difference between a scalar matrix and a diagonal matrix?

A Scalar matrix has zeros on diagonal

B Diagonal matrix must have equal elements

C Scalar matrix has equal diagonal elements; diagonal matrix can have different ones

D No difference

💡 Short Explanation

Scalar matrices are special diagonal matrices. Scalar matrices require equal diagonal entries. Diagonal matrices allow unequal entries.

✅ Correct Answer Explanation

A diagonal matrix only requires all non-diagonal elements to be zero. A scalar matrix additionally requires all diagonal elements to be equal. Thus scalar matrices form a special category within diagonal matrices. Therefore Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Scalar matrices generally have non-zero diagonal entries.
Option B → Diagonal matrices need not have equal diagonals.
Option D → Scalar and diagonal matrices are related but not identical.

🎯 Strategy

Used: Odd One Out

Application:

Identify the extra condition defining scalar matrices.

Final Logic:

Equal diagonal entries distinguish scalar matrices.

🧠 Memory Trick

"Scalar = same diagonal"

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

UG Mathematics Booster Test 3 - Types of Matrices

Test Complete!

Answer Review