CUET UG Applied Mathematics Booster Test 2 - Matrix Fundamentals and Types -

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CUET UG Applied Mathematics Booster Test 2 - Matrix Fundamentals and Types

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 2 - Matrix Fundamentals and Types

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QUESTION 1 OF 20

If an orientation of data is changed from matrix A of order 2 × 3 to matrix B of order 3 × 2, does the fundamental definition of the matrix as a representation of m rows and n columns change the total number of elements?

Answer Review

1 If an orientation of data is changed from matrix A of order 2 × 3 to matrix B of order 3 × 2, does the fundamental definition of the matrix as a representation of m rows and n columns change the total number of elements?

A Yes, because the orientation alters the total structural values.

B No, the total number of elements remains mn.

C Yes, because square matrices have different elements.

D No, because rows and columns are always equal.

💡 Short Explanation

�� Total elements in a matrix = m × n �� 2 × 3 and 3 × 2 both contain 6 elements �� Orientation changes arrangement, not quantity

✅ Correct Answer Explanation

The order of a matrix is represented as m × n, where m is the number of rows and n is the number of columns. → Total elements are always obtained by multiplying rows and columns. For matrix A of order 2 × 3: 2 × 3 = 6 elements For matrix B of order 3 × 2: 3 × 2 = 6 elements Hence, only the arrangement changes, not the total number of entries. → Option A is incorrect because changing orientation does not alter the total count. → Option B is correct because mn remains constant. → Option C is incorrect because the matrices are not square matrices. → Option D is incorrect because rows and columns are not necessarily equal in every matrix.

❓ Why Other Options Are Incorrect

�� Option A → Orientation changes placement of entries, not the number of entries.
�� Option C → Square matrices are unrelated to this comparison.
�� Option D → Rows and columns are equal only in square matrices.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Compute total elements using rows × columns for both matrices.

Final Logic: Since 2 × 3 = 3 × 2 = 6, the number of elements remains unchanged.

🧠 Memory Trick

"Rows × Columns = Total Entries"

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

2 In matrix A = [aᵢⱼ], if a₁₁ = 2 + i, a₂₁ = -1, a₃₁ = 4 - 2i, and a₃₂ = 9, what can be inferred analytically about the acceptable elements of the matrix?

A Elements must strictly be real integers.

B Elements can be complex numbers like 2 + i.

C The matrix only has 1 column.

D All diagonal elements must be real.

💡 Short Explanation

�� Matrix elements may be real or complex �� 2 + i is a complex number �� Matrices are not restricted to integers only

✅ Correct Answer Explanation

Matrix entries can belong to different number systems including real numbers and complex numbers. Given: a₁₁ = 2 + i and a₃₁ = 4 − 2i These are complex numbers because they contain the imaginary unit i. Therefore, matrices can contain complex-valued entries. → Option A is incorrect because matrices are not restricted to integers. → Option B is correct since complex numbers are valid matrix elements. → Option C is incorrect because the existence of a₃₂ proves there are at least 2 columns. → Option D is incorrect because diagonal elements can also be complex.

❓ Why Other Options Are Incorrect

�� Option A → Matrix elements need not be integers only.
�� Option C → a₃₂ indicates second column existence.
�� Option D → Diagonal entries may also be complex numbers.

🎯 Strategy

Used: Elimination

Application: Presence of imaginary unit directly removes integer-only assumptions.

Final Logic: Since 2 + i is a valid matrix entry, complex elements are allowed.

🧠 Memory Trick

"Matrices accept real + imaginary entries."

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

3 If a matrix has an integral number of elements equal to exactly a prime number p, what are the possible orders of this matrix?

A p × p

B 1 × p and p × 1

C 2 × p

D 1 × 1

💡 Short Explanation

�� Number of elements = Rows × Columns. �� A prime number has only two positive factors. �� Therefore, the possible orders are 1 × p and p × 1.

✅ Correct Answer Explanation

If a matrix has m rows and n columns, then the total number of elements is: Number of elements = m × n → Since the total number of elements is a prime number p, its only positive factors are 1 and p. → Therefore, the only possible matrix orders are: 1 × p or p × 1 → Option B is correct because both orders contain exactly p elements. → Option A gives: p × p = p² elements, which is greater than p. → Option C gives: 2 × p = 2p elements, which is not equal to p. → Option D gives: 1 × 1 = 1 element, which is not equal to p (as every prime number is greater than 1).

❓ Why Other Options Are Incorrect

�� Option A → Produces p² elements instead of p elements.
�� Option C → Produces 2p elements instead of p elements.
�� Option D → Produces only one element, whereas a prime number is greater than 1.

🎯 Strategy

Used

Substitution

Application:

�� Express the total number of elements as m × n and use the factorization property of prime numbers to determine the possible matrix orders.

Final Logic:

�� Since a prime number has only two positive factors, 1 and itself, the only possible matrix orders are 1 × p and p × 1.

🧠 Memory Trick

"Prime = 1 × p only."

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Order of a Matrix and Elements of a Matrix

4 When expressing a coordinate area using general notation A = [aᵢⱼ]ₘₓₙ, what does the suffix 'j' distinctly identify?

A The row in which the element is located

B The column in which the element is located

C The total number of rows

D The diagonal parameter

💡 Short Explanation

�� i denotes row index �� j denotes column index �� aᵢⱼ identifies matrix position

✅ Correct Answer Explanation

In standard matrix notation aᵢⱼ: • i represents row number • j represents column number Thus, suffix j specifically identifies the column location of the element. → Option A is incorrect because rows are identified by i. → Option B is correct. → Option C is incorrect because total rows are represented by m. → Option D is incorrect because j has no relation with diagonal property directly.

❓ Why Other Options Are Incorrect

�� Option A → Row index corresponds to i, not j.
�� Option C → Total rows are represented by m.
�� Option D → j only denotes column position.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match standard notation meanings directly.

Final Logic: In aᵢⱼ, j always represents column number.

🧠 Memory Trick

"j → vertical column jump."

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: General Form of a Matrix

5 Can a single vector column matrix be classified as a rectangular matrix?

A Yes, because m ≠ n for any column matrix where m > 1.

B No, because it is a single vector matrix, not rectangular.

C Yes, but only if m = 1.

D No, column matrices are universally square matrices.

💡 Short Explanation

�� Rectangular matrices have m ≠ n �� Column matrix has order m × 1 �� If m > 1, then m ≠ 1

✅ Correct Answer Explanation

A rectangular matrix is any matrix where the number of rows and columns are unequal. A column matrix has order m × 1. If m > 1, then m ≠ n. Hence, it becomes a rectangular matrix. → Option A is correct. → Option B is incorrect because a column matrix may also be rectangular. → Option C is incorrect because m = 1 creates a 1 × 1 square matrix. → Option D is incorrect because column matrices are not always square.

❓ Why Other Options Are Incorrect

�� Option B → Column matrices can also satisfy rectangular conditions.
�� Option C → 1 × 1 is square, not rectangular.
�� Option D → Only 1 × 1 column matrices are square.

🎯 Strategy

Used: Elimination

Application: Compare definition of rectangular matrix with column matrix order.

Final Logic: Since m ≠ 1 for most column matrices, they are rectangular.

🧠 Memory Trick

"Unequal rows and columns ⇒ rectangular."

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

6 If a matrix is constructed randomly with order m × n where 1 ≤ m ≤ 3 and 1 ≤ n ≤ 3, what is the probability that the resulting format is a Square Matrix?

A 1/9

B 1/3

C 1/2

D 3/4

💡 Short Explanation

�� Possible row choices = 3 �� Possible column choices = 3 �� Square matrices occur when m = n

✅ Correct Answer Explanation

Possible matrix orders are: 1×1, 1×2, 1×3, 2×1, 2×2, 2×3, 3×1, 3×2, 3×3 Total possibilities = 9 Square matrices are: 1×1, 2×2, 3×3 Number of square matrices = 3 Probability = 3/9 = 1/3 → Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Counts only one favorable case.
�� Option C → Incorrect favorable-total ratio.
�� Option D → Too large compared to actual count.

🎯 Strategy

Used: Substitution

Application: List all possible orders and count square matrices.

Final Logic: 3 square cases out of 9 total cases gives 1/3.

🧠 Memory Trick

"Square means rows = columns."

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

7 A moving average sequence is stored in a Row matrix of order 1 × 4. If transposed, what will be the order of the resulting matrix?

A 4 × 4

B 1 × 4

C 4 × 1

D 1 × 1

💡 Short Explanation

�� Transpose interchanges rows and columns �� 1 × 4 becomes 4 × 1 �� Row matrix converts to column matrix

✅ Correct Answer Explanation

The transpose of a matrix changes rows into columns and columns into rows. Given order = 1 × 4 After transpose: 4 × 1 → Option C is correct. → Option A is incorrect because transpose does not create square matrices automatically. → Option B is incorrect because order changes after transpose. → Option D is incorrect because total elements remain 4.

❓ Why Other Options Are Incorrect

�� Option A → Transpose does not increase dimensions.
�� Option B → Order changes after transpose.
�� Option D → Matrix still contains four elements.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Reverse row-column positions directly.

Final Logic: Transpose of 1 × 4 is 4 × 1.

🧠 Memory Trick

"Transpose flips dimensions."

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

8 In an EMI calculation represented by a Column Matrix C of order 3 × 1, what strictly defines its structure?

A It has exactly one row for 3 columns.

B It has exactly one column for 3 rows.

C It is a symmetric square matrix.

D Its non-diagonal elements are non-zero.

💡 Short Explanation

�� Column matrix has one column �� Order 3 × 1 means 3 rows and 1 column �� It is not square

✅ Correct Answer Explanation

Matrix order m × n means: m = rows n = columns Thus, 3 × 1 means: • 3 rows • 1 column Hence, it is a column matrix. → Option B is correct. → Option A reverses rows and columns incorrectly. → Option C is wrong because 3 × 1 is not square. → Option D is unrelated to column matrix definition.

❓ Why Other Options Are Incorrect

�� Option A → Incorrect interpretation of order notation.
�� Option C → Symmetric matrices must be square.
�� Option D → Non-diagonal conditions do not define column matrices.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match order notation directly with matrix structure.

Final Logic: 3 × 1 always means 3 rows and 1 column.

🧠 Memory Trick

"First number = rows, second = columns."

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

9 What is the exact constraint on the elements aᵢⱼ for a matrix to fall into the region of Zero Matrices?

A aᵢⱼ = 1 for all i, j

B aᵢⱼ = 0 for all i ≠ j

C aᵢⱼ = 0 for all i, j

D aᵢⱼ = i + j

💡 Short Explanation

�� Every element must be zero �� Applies to all positions �� Zero matrix may have any order

✅ Correct Answer Explanation

A zero matrix is defined as a matrix in which every element equals zero. Mathematically: aᵢⱼ = 0 for all i and j → Option A is incorrect because all entries are 1, not 0. → Option B describes diagonal matrices with zero off-diagonal entries. → Option C is correct. → Option D is unrelated.

❓ Why Other Options Are Incorrect

�� Option A → Represents all-one matrices, not zero matrices.
�� Option B → Only off-diagonal elements are zero there.
�� Option D → Gives varying non-zero entries.

🎯 Strategy

Used: Elimination

Application: Apply exact definition of zero matrix.

Final Logic: Every entry must be zero in a zero matrix.

🧠 Memory Trick

"Zero matrix ⇒ zero everywhere."

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

10 A mixture equation gives equal matrices [2a+b a-2b] = [4 -3]. What is the value of 'a' and 'b' derived from this equality?

A a = 1, b = 2

B a = 2, b = 1

C a = -1, b = -2

D a = 3, b = 4

💡 Short Explanation

�� Equal matrices have equal corresponding entries �� Solve simultaneous equations �� Compare positions directly

✅ Correct Answer Explanation

For equal matrices, corresponding elements are equal. Thus: 2a + b = 4 a − 2b = −3 From second equation: a = 2b − 3 Substitute into first: 2(2b − 3) + b = 4 4b − 6 + b = 4 5b = 10 b = 2 Then: a = 2(2) − 3 = 1 Hence: a = 1, b = 2 → Option A is correct.

❓ Why Other Options Are Incorrect

�� Option B → Does not satisfy second equation.
�� Option C → Fails both equations.
�� Option D → Produces incorrect matrix entries.

🎯 Strategy

Used: Substitution

Application: Equate corresponding entries and solve simultaneously.

Final Logic: Solving the equations gives a = 1 and b = 2.

🧠 Memory Trick

"Equal matrices ⇒ equal positions."

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

11 Identify the incorrect statement regarding diagonal elements of a matrix:

A They exist only where the row index equals the column index (i = j).

B They are denoted by a₁₁, a₂₂, a₃₃, etc.

C They can form a main diagonal in a purely rectangular matrix (non-square matrix).

D A matrix where all non-diagonal elements are zero is called a diagonal matrix.

💡 Short Explanation

�� Diagonal elements are defined only for square matrices. �� In a rectangular matrix, a complete principal diagonal is not properly defined. �� A diagonal matrix must be square with non-diagonal entries zero.

✅ Correct Answer Explanation

Diagonal elements occur where the row index equals the column index, i.e., i=j. Hence statements A and B are correct. → A diagonal matrix is a square matrix in which all non-diagonal elements are zero, so statement D is also correct. → Statement C is incorrect because a proper main diagonal is fundamentally associated with square matrices. Rectangular matrices do not satisfy the structural requirement for defining a complete diagonal matrix concept.

❓ Why Other Options Are Incorrect

�� Option A → Correctly defines diagonal elements using the condition i=j.
�� Option B → Correct notation for diagonal elements in matrices.
�� Option D → This is the standard definition of a diagonal matrix.

🎯 Strategy

Used: Elimination

Application: Eliminate statements that match standard matrix definitions from NCERT terminology.

Final Logic: Since diagonal matrices must be square matrices, the rectangular matrix statement is incorrect.

🧠 Memory Trick

"Diagonal ⇒ Square only."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Diagonal Elements and Diagonal Matrix

12 Which of the following analytical statements about non-diagonal elements are correct?
1. They are elements aᵢⱼ where i ≠ j.
2. In a diagonal matrix, all non-diagonal elements are explicitly zero.
3. In a scalar matrix, non-diagonal elements can be non-zero.
4. In an identity matrix, all non-diagonal elements are explicitly zero.

A 1,2,4

B 1,2

C 1,2,3

D 1,2,3,4

💡 Short Explanation

�� Non-diagonal elements satisfy i≠j. �� Diagonal and identity matrices have zero non-diagonal entries. �� Scalar matrices also have zero non-diagonal elements.

✅ Correct Answer Explanation

Statement 1 is correct because non-diagonal elements are defined by i≠j. → Statement 2 is correct since diagonal matrices contain zeros everywhere except possibly on the diagonal. → Statement 3 is incorrect because scalar matrices are special diagonal matrices; hence all non-diagonal elements must also be zero. → Statement 4 is correct because an identity matrix has 1s on the diagonal and 0s elsewhere.

❓ Why Other Options Are Incorrect

�� Option B → Omits statement 4, which is correct.
�� Option C → Includes statement 3, which is false for scalar matrices.
�� Option D → Incorrect because statement 3 is false.

🎯 Strategy

Used: Option Grouping

Application: Check which statements align with properties shared by diagonal, scalar, and identity matrices.

Final Logic: Since scalar matrices also require zero non-diagonal entries, only A, B, and D remain correct.

🧠 Memory Trick

"Diagonal family → outside entries zero."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Diagonal, Scalar and Identity Matrices

13 Match the matrix type to its specific non-diagonal constraint condition:

List I	List II
1. Diagonal Matrix	a. aᵢⱼ = 0 for i ≠ j, and aᵢᵢ = 1
2. Scalar Matrix	b. aᵢⱼ = 0 for all i, j
3. Identity Matrix	c. aᵢⱼ = 0 for i ≠ j, and aᵢᵢ = k (where k is same)
4. Zero Matrix	d. aᵢⱼ = 0 for i ≠ j

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

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

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

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

💡 Short Explanation

�� Diagonal matrices have zero non-diagonal entries. �� Scalar matrices have equal diagonal entries. �� Identity matrices have diagonal entries equal to 1.

✅ Correct Answer Explanation

Diagonal Matrix corresponds to condition (IV), where all non-diagonal elements are zero. → Scalar Matrix corresponds to (III), where diagonal entries are equal to constant k. → Identity Matrix corresponds to (I), where diagonal entries are 1 and non-diagonal entries are zero. → Zero Matrix corresponds to (II), where every element is zero.

❓ Why Other Options Are Incorrect

�� Option B → Incorrectly interchanges scalar and identity matrix conditions.
�� Option C → Wrongly maps scalar matrix with identity matrix property.
�� Option D → Diagonal matrix does not require equal diagonal entries.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match each matrix type with its defining property carefully.

Final Logic: Identity matrices require 1s on diagonal, scalar matrices require equal constants.

🧠 Memory Trick

"Identity → ones, Scalar → same constant."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Types of Matrices

14 If matrix B is a scalar matrix of order 3, what must be true about its diagonal elements?

A They are all equal to 0.

B They are all distinct numbers.

C They are all equal to some constant k.

D They must equal to 1.

💡 Short Explanation

�� Scalar matrices are special diagonal matrices. �� All diagonal entries are equal. �� Non-diagonal entries are zero.

✅ Correct Answer Explanation

A scalar matrix is defined as a diagonal matrix in which all diagonal entries are equal to the same constant k. → Option A describes a zero matrix only if all entries are zero. → Option D describes an identity matrix where k=1. → Hence Option C correctly defines the scalar matrix property.

❓ Why Other Options Are Incorrect

�� Option A → Scalar matrices need not always have zero diagonal entries.
�� Option B → Diagonal entries cannot be distinct in a scalar matrix.
�� Option D → Only identity matrices have all diagonal entries equal to 1.

🎯 Strategy

Used: Elimination

Application: Remove options representing special cases like zero and identity matrices.

Final Logic: Scalar matrices require identical diagonal entries equal to constant k.

🧠 Memory Trick

"Scalar = Same diagonal scalar."

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

15 Is every identity matrix also technically a scalar matrix?

A No, because identity matrices must have 1s, scalar can be anything.

B Yes, because an identity matrix is a scalar matrix where k = 1.

C Yes, but only for 2 × 2 matrices.

D No, identity matrices are intrinsically rectangular.

💡 Short Explanation

�� Identity matrix is a special scalar matrix. �� Diagonal entries are all 1. �� Non-diagonal entries are zero.

✅ Correct Answer Explanation

A scalar matrix has equal diagonal entries k and zero non-diagonal entries. → If k=1, the scalar matrix becomes an identity matrix. → Therefore every identity matrix is a scalar matrix, but not every scalar matrix is an identity matrix.

❓ Why Other Options Are Incorrect

�� Option A → Identity matrices satisfy the scalar matrix condition with k=1.
�� Option C → The definition applies to matrices of every order, not only 2×2.
�� Option D → Identity matrices are square matrices, not rectangular.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Relate the definition of identity matrix to scalar matrix conditions.

Final Logic: Identity matrix is simply a scalar matrix with scalar value 1.

🧠 Memory Trick

"Identity = Scalar with k = 1."

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

16 For a unit matrix Iₙ, what does the subscript 'n' designate analytically?

A The number of non-diagonal elements

B The scalar multiple factor

C The order of the square matrix

D The total number of elements

💡 Short Explanation

�� I_n denotes identity matrix of order n. �� Identity matrices are square matrices. �� Number of rows equals number of columns.

✅ Correct Answer Explanation

In notation I_n, the subscript n specifies that the matrix has order n×n. → Since identity matrices are square matrices, both rows and columns equal n. → Thus Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Non-diagonal elements are not represented by the subscript.
�� Option B → No scalar multiplication is implied.
�� Option D → Total elements would be n^2, not n.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Interpret standard notation used in matrix algebra.

Final Logic: I_n directly indicates an identity matrix of order n.

🧠 Memory Trick

"Subscript tells size."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Identity Matrix Notation

17 How many distinct 2 × 2 matrices can be formed using the numbers 5, 7, and -1, allowing repetition?

A 2³

B 3⁴

C 4³

D 3 × 4

💡 Short Explanation

�� A 2×2 matrix has 4 entries. �� Each entry has 3 choices. �� Repetition is allowed.

✅ Correct Answer Explanation

A 2×2 matrix contains 4 positions. → Each position can independently take any of the 3 values: 5, 7, or -1. → Therefore total matrices =3×3×3×3=3^4.

❓ Why Other Options Are Incorrect

�� Option A → Uses only 3 positions instead of 4.
�� Option C → Incorrect number of choices.
�� Option D → Does not apply permutation or counting principle properly.

🎯 Strategy

Used: Substitution

Application: Apply counting principle directly to each matrix entry.

Final Logic: Four entries with 3 choices each gives 3^4.

🧠 Memory Trick

"Entries count powers."

📚 Class XI – NCERT Mathematics – Chapter: Permutations and Combinations – Topic: Fundamental Principle of Counting

18 Construct matrix B = [bᵢⱼ] of order 2 × 2 where bᵢⱼ = (i + 2j)² / 2. What is the calculated element b₂₁?

A 8

B 9/2

C 25/2

D 18

💡 Short Explanation

�� Use formula b_(ij)=(i+2j)^2/2. �� Substitute i=2,j=1. �� Simplify carefully.

✅ Correct Answer Explanation

For element b_(21), substitute i=2 and j=1: b_(21)=(2+2⋅1)^2/2=4^2/2=16/2=8 → Therefore the correct value is 8.

❓ Why Other Options Are Incorrect

�� Option B → Incorrect squaring or division step.
�� Option C → Comes from wrong substitution values.
�� Option D → Arithmetic simplification error.

🎯 Strategy

Used: Substitution

Application: Substitute row and column indices directly into the formula.

Final Logic: Correct substitution gives 8.

🧠 Memory Trick

"Plug indices, then square."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Construction of Matrices

📄 Passage

Data processing often requires converting a tabular dataset into an ordered matrix to successfully apply mathematical operations. When you possess two matrices A and B of the exact same order, say m × n, and k and p are constant scalars, algebraic properties hold true. For instance, operations such as k(A + B) = kA + kB and (k + p)A = kA + pA allow for simultaneous scaling of data. Suppose a school collects tabular data representing the number of boys and girls in three different distinct groups. This data naturally forms a rectangular matrix of order 3 × 2. By clearly understanding the coordinate position aᵢⱼ of each specific data point, such as a₃₂ representing girls in the third group, transformations become easier. Scalar multipliers can then uniformly update the entire system without losing the structural integrity of the original coordinate system.

Question:
If the school uniformly doubles the number of boys and girls in all three groups represented in the tabular matrix A, what property of scalar multiplication is applied?

A Negative matrix conversion

B Matrix addition

C kA where k = 2

D Coordinate interchange

💡 Short Explanation

�� Scalar multiplication multiplies every entry by a constant. �� Doubling means multiplying by 2. �� Matrix structure remains unchanged.

✅ Correct Answer Explanation

Uniformly doubling all entries means multiplying the matrix by scalar 2. → If matrix is A, then updated matrix becomes 2A. → This operation is called scalar multiplication of matrices.

❓ Why Other Options Are Incorrect

�� Option A → No negative transformation occurs.
�� Option B → Addition is not involved here.
�� Option D → Coordinates are not interchanged.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match the phrase "uniformly doubles" with scalar multiplication.

Final Logic: Multiplying every matrix entry by 2 gives 2A.

🧠 Memory Trick

"Double ⇒ multiply by scalar 2."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Scalar Multiplication

📄 Passage

Question:
In the given context, what does the specific coordinate element a₃₂ signify in the 3 × 2 matrix of student groups?

A Boys in the second group

B Girls in the third group

C Total students in the third group

D Girls in the second group

💡 Short Explanation

�� First index denotes row. �� Second index denotes column. �� a_(32)means row 3, column 2.

✅ Correct Answer Explanation

In matrix notation a_(ij), i represents the row number and j represents the column number. → Here rows correspond to student groups and columns correspond to boys and girls. → Thus a_(32)refers to the entry in the 3rd row and 2nd column, i.e., girls in the third group.

❓ Why Other Options Are Incorrect

�� Option A → Refers to second group data, not third row.
�� Option C → a_(32)represents a single entry, not total students.
�� Option D → Refers to second group instead of third group.

🎯 Strategy

Used: Substitution

Application: Decode indices systematically as row-column positions.

Final Logic: Third row and second column correspond to girls in the third group.

🧠 Memory Trick

"First index → row, second → column."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Element of a Matrix

CUET UG Applied Mathematics Booster Test 2 – Matrix Fundamentals and Types

CUET UG Applied Mathematics Booster Test 2 - Matrix Fundamentals and Types

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