UG Mathematics Booster Test 2 - Introduction to Matrices -

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UG Mathematics Booster Test 2 - Introduction to Matrices

⏱️ 30 Minutes

❓ 20 Questions

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

UG Mathematics Booster Test 2 - Introduction to Matrices

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QUESTION 1 OF 20

What are the individual numbers or functions contained inside the ordered rectangular array officially called?

Answer Review

1 What are the individual numbers or functions contained inside the ordered rectangular array officially called?

A Determinants

B Constants

C Variables

D Elements or entries

💡 Short Explanation

Matrix contents are called entries. Each position contains an element. Entries may be numbers or functions.

✅ Correct Answer Explanation

The individual quantities present inside a matrix are officially called elements or entries. These entries occupy specific row-column positions. Hence Option D is correct. Determinants are separate scalar values associated with square matrices. Constants and variables may appear inside matrices, but they are not the general name for matrix components.

❓ Why Other Options Are Incorrect

Option A → Determinants are numerical values associated only with square matrices and are not the entries themselves.
Option B → Constants are fixed numerical quantities, but matrix entries need not always be constants.
Option C → Variables may occur in matrices, but "variables" is not the standard term for all matrix components.

🎯 Strategy

Used: Elimination

Application:

Identify the official mathematical terminology used for quantities inside matrices.

Final Logic:

"Elements or entries" is the standard NCERT term for matrix contents.

🧠 Memory Trick

"Matrix entries = Elements"

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

2 Which of the following is NOT listed in the text as a practical, real-life use of matrices?

A Industrial management

B Cost estimation and budgeting

C Measuring celestial distances

D Cryptography and modern psychology

💡 Short Explanation

Matrices are used in management and cryptography. Budgeting applications are common. Celestial distance measurement is not listed.

✅ Correct Answer Explanation

NCERT discusses applications of matrices in industrial management, economics, psychology, cryptography, and data organization. Measuring celestial distances is not specifically mentioned among standard matrix applications in the text. Therefore Option C is correct. Options A, B, and D all reflect recognized practical uses of matrices.

❓ Why Other Options Are Incorrect

Option A → Industrial management commonly uses matrices for resource allocation and planning.
Option B → Cost estimation and budgeting involve tabular numerical analysis using matrices.
Option D → Cryptography and psychology are established practical application areas of matrices.

🎯 Strategy

Used: Odd One Out

Application:

Identify the option not matching the commonly listed NCERT applications of matrices.

Final Logic:

Only celestial distance measurement is not directly listed in the chapter context.

🧠 Memory Trick

"Matrices manage data, not stars."

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

3 Assertion (A): The evolution of the concept of matrices was a direct result of an attempt to obtain compact methods of solving systems of linear equations.
Reason (R): Matrix notation significantly complicates the mathematical method of solving systems of linear equations.

A Both False

B A True, R False

C Both True, R explains A

D A False, R True

💡 Short Explanation

Matrices simplify linear equations. Compact representation motivated development. Matrix notation does not complicate solving.

✅ Correct Answer Explanation

The Assertion is true because matrices developed mainly to simplify and compactly solve systems of linear equations. The Reason is false because matrix notation actually simplifies calculations and organization. Hence Option B is correct. Matrices provide efficient algebraic representation rather than creating mathematical complications.

❓ Why Other Options Are Incorrect

Option A → The Assertion is historically and mathematically correct.
Option C → The Reason is false because matrices simplify rather than complicate calculations.
Option D → The Assertion is true, so this option is incorrect.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Verify Assertion and Reason independently and test logical compatibility.

Final Logic:

Matrices simplify equation solving; they do not complicate it.

🧠 Memory Trick

"Matrix = Compact Solver"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Origin and Purpose of Matrices

4 Arrange the computational steps to construct a 2×2 numerical matrix where elements are defined by the rule a_ij = i + j:
1. Calculate a_11 = 2
2. Calculate a_12 = 3
3. Calculate a_21 = 3
4. Calculate a_22 = 4

A 1-2-3-4

B 4-3-2-1

C 2-1-4-3

D 3-4-1-2

💡 Short Explanation

Elements are filled systematically. Indices follow row-column order. Matrix entries proceed sequentially.

✅ Correct Answer Explanation

For a 2×2 matrix, entries are calculated row-wise using the rule aᵢⱼ = i + j. First calculate a₁₁, then a₁₂, followed by a₂₁ and a₂₂. Therefore the correct sequence is 1-2-3-4. Other options disturb the standard row-column arrangement used in matrix construction.

❓ Why Other Options Are Incorrect

Option B → Reverses the standard matrix filling sequence completely.
Option C → Disrupts the natural row-wise arrangement of entries.
Option D → Starts from second-row elements before completing the first row.

🎯 Strategy

Used: Option Grouping

Application:

Follow standard row-column indexing rules while constructing the matrix.

Final Logic:

Matrices are normally filled sequentially from a₁₁ onward.

🧠 Memory Trick

"First row first, second row later."

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Matrix Formation Using Rules

5 A business spreadsheet forms a matrix with exactly 13 numerical elements. How many possible valid analytical orders (m × n) can this matrix have?

A 1

B 4

C 2

D 3

💡 Short Explanation

Total elements = m × n. 13 is a prime number. Possible orders are 1×13 and 13×1.

✅ Correct Answer Explanation

The order of a matrix depends on factor pairs of the total number of elements. Since 13 is prime, it has only two factor pairs: \(13=1\times 13=13\times 1\) Thus only two valid matrix orders are possible: 1×13 and 13×1. Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → There are two possible arrangements, not one.
Option B → Four orders would require more factor pairs than a prime number provides.
Option D → Prime numbers do not produce three distinct factor-based matrix orders.

🎯 Strategy

Used: Substitution

Application:

Determine factor pairs of 13 to identify possible matrix orders.

Final Logic:

Prime number 13 gives exactly two matrix orders.

🧠 Memory Trick

"Prime → Only two orders"

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

6 In a scientific experiment, data is represented by a 2×2 matrix with elements 2, 4, 6, and 8. What is the mean (average) of all elements?

A 8

B 9

C 10

D 5

💡 Short Explanation

Mean = Total sum ÷ Number of elements. Sum = 20. Average = 20 ÷ 4 = 5.

✅ Correct Answer Explanation

The average of matrix elements is calculated by adding all entries and dividing by the number of entries: \(\frac{2+4+6+8}{4}=5\) Thus the mean value is 5, making Option D correct. Other options are either individual entries or incorrect arithmetic results.

❓ Why Other Options Are Incorrect

Option A → 8 is only the largest matrix entry, not the mean.
Option B → 9 exceeds the average possible from the given data.
Option C → 10 equals half the total sum, not the average.

🎯 Strategy

Used: Substitution

Application:

Substitute matrix values into the arithmetic mean formula.

Final Logic:

Total 20 divided by 4 gives mean 5.

🧠 Memory Trick

"Mean = Sum ÷ Count"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Numerical Data Representation

7 If a single geometric point is represented strictly as a column matrix, what is its order?

A 2×1

B 1×2

C 2×2

D 1×1

💡 Short Explanation

A point has x and y coordinates. Column matrices stack values vertically. Two rows and one column are formed.

✅ Correct Answer Explanation

A geometric point (x, y) represented as a column matrix is written vertically: \(\left[\begin{aligned}x\\ y\end{aligned}\right]\) This matrix has 2 rows and 1 column, so its order is 2×1. Hence Option A is correct. Other options represent row matrices or incorrect dimensions.

❓ Why Other Options Are Incorrect

Option B → 1×2 represents a row matrix, not a column matrix.
Option C → A 2×2 matrix requires four elements, not two coordinates.
Option D → A 1×1 matrix contains only one entry.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Identify rows and columns based on vertical arrangement.

Final Logic:

Two vertically arranged coordinates produce a 2×1 matrix.

🧠 Memory Trick

"Column means vertical stack."

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

8 A closed figure has 4 vertices. Representing these as a 2×4 matrix, how many total entries are there?

A 4

B 8

C 6

D 12

💡 Short Explanation

Total entries = rows × columns. Matrix order is 2×4. Total entries equal 8.

✅ Correct Answer Explanation

The total number of entries in a matrix equals the product of rows and columns: \(2\times 4=8\) Thus a 2×4 matrix contains 8 entries. Therefore Option B is correct. The other options do not match the standard formula for counting matrix elements.

❓ Why Other Options Are Incorrect

Option A → 4 counts only columns, ignoring rows.
Option C → 6 does not equal the product of 2 and 4.
Option D → 12 would correspond to a larger matrix size.

🎯 Strategy

Used: Substitution

Application:

Multiply rows and columns directly to determine entries.

Final Logic:

2 × 4 gives exactly 8 elements.

🧠 Memory Trick

"Entries = Rows × Columns"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Order and Elements

9 Which characteristics define a 3×2 matrix?
1. It has exactly 2 horizontal rows
2. It has exactly 3 horizontal rows
3. It has exactly 2 vertical columns

A 1 and 3 only

B 2 and 3 only

C 1 and 2 only

D 1, 2, and 3

💡 Short Explanation

First number denotes rows. Second number denotes columns. 3×2 means 3 rows and 2 columns.

✅ Correct Answer Explanation

In a matrix of order m×n, m represents rows and n represents columns. Therefore a 3×2 matrix has 3 horizontal rows and 2 vertical columns. Statement 2 and Statement 3 are correct, making Option B correct. Statement 1 is incorrect because the matrix does not have 2 rows.

❓ Why Other Options Are Incorrect

Option A → Statement 1 is false because rows are actually 3.
Option C → Statement 3 is also true and cannot be omitted.
Option D → Including Statement 1 makes the option incorrect.

🎯 Strategy

Used: Elimination

Application:

Interpret matrix order carefully using row-column notation.

Final Logic:

3×2 means 3 rows and 2 columns only.

🧠 Memory Trick

"First rows, second columns."

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

10 Match the matrix structure to its mathematical definition:

List I	List II
1. Column matrix	a. m = n
2. Row matrix	b. m × 1
3. Square matrix	c. 1 × n

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

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

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

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

💡 Short Explanation

Column matrix has one column. Row matrix has one row. Square matrix has equal rows and columns.

✅ Correct Answer Explanation

A column matrix has order m×1, a row matrix has order 1×n, and a square matrix satisfies m = n. Therefore the correct matching is 1-b, 2-c, 3-a. Option A correctly pairs all definitions according to standard NCERT matrix terminology.

❓ Why Other Options Are Incorrect

Option B → Column and row matrix definitions are interchanged incorrectly.
Option C → Square matrices are defined by equal rows and columns, not 1×n.
Option D → Row matrices are not characterized by m = n.

🎯 Strategy

Used: Option Grouping

Application:

Match each matrix type with its defining dimensional property.

Final Logic:

Only Option A correctly matches all matrix structures.

🧠 Memory Trick

"Column → one column, Row → one row"

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

11 Two matrices enclosed in standard brackets A = [a_ij] and B = [b_ij] are defined as algebraically equal ONLY IF:

A They have the same number of elements but different orders.

B They are both square matrices of any order.

C They have the same number of rows but different columns.

D They have the same order and their corresponding elements are exactly equal.

💡 Short Explanation

Equal matrices must match completely. Orders must be identical. Corresponding entries must also be equal.

✅ Correct Answer Explanation

Two matrices are equal only when they have the same order and every corresponding element is equal. Therefore Option D correctly states the condition for matrix equality. Option A ignores order equality. Option B is insufficient because square matrices of different orders are not equal. Option C fails because unequal columns produce different orders.

❓ Why Other Options Are Incorrect

Option A → Same number of elements alone does not guarantee equality if matrix orders differ.
Option B → Merely being square matrices is insufficient; corresponding elements must also match.
Option C → Different columns imply different orders, so matrices cannot be equal.

🎯 Strategy

Used: Elimination

Application:

Apply the formal definition of equality of matrices directly to remove incomplete conditions.

Final Logic:

Equal matrices require both equal order and equal corresponding entries.

🧠 Memory Trick

"Same size + same entries = equal matrices"

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

12 If a matrix element is randomly selected from a valid real-number matrix, what is the probability that the element results in a real value?

A 50%

B 0%

C 100% (or 1)

D 25%

💡 Short Explanation

Real-number matrices contain only real entries. Every selected element is real. Probability therefore equals 1.

✅ Correct Answer Explanation

A real-number matrix contains only real-valued elements. Therefore any randomly selected entry from such a matrix must also be real. Hence the probability is 100% or 1, making Option C correct. The other probabilities incorrectly assume the possibility of non-real entries in a real matrix.

❓ Why Other Options Are Incorrect

Option A → There is no 50% uncertainty because all entries are real numbers.
Option B → Probability cannot be zero since every matrix element is real.
Option D → A 25% probability has no mathematical basis here.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Interpret the meaning of "real-number matrix" carefully before evaluating probability.

Final Logic:

All entries are real, so probability equals 1.

🧠 Memory Trick

"Real matrix → Real entries only"

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

13 A 1 x 2 row matrix has its first numerical element defined by the integral of 3x² dx from 0 to 1. What is the exact value of this element?

A 1

B 0

C 2

D 3

💡 Short Explanation

Integrate 3x² carefully. Antiderivative becomes x³. Substituting limits gives 1.

✅ Correct Answer Explanation

The definite integral is evaluated as: \(\int_{0}^{1}\,3x^{2} dx={\left[x^{3}\right]}_{0}^{1}=1\) Hence the first matrix element equals 1, making Option A correct. The other options result from incorrect integration or limit substitution.

❓ Why Other Options Are Incorrect

Option B → The integral over a positive interval cannot equal zero here.
Option C → Incorrect integration or substitution leads to this value.
Option D → 3 represents the coefficient, not the definite integral result.

🎯 Strategy

Used: Substitution

Application:

Integrate first, then apply upper and lower limits systematically.

Final Logic:

1³ − 0³ gives exactly 1.

🧠 Memory Trick

"3x² integrates to x³"

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

14 A column matrix holds the physical length and width of a rectangle numerically as 6 and 4 respectively. What is the calculated area?

A 10

B 24

C 20

D 2

💡 Short Explanation

Rectangle area = length × width. Values are 6 and 4. Product equals 24.

✅ Correct Answer Explanation

The dimensions stored in the column matrix represent rectangle length and width. Therefore area is calculated by multiplication: \(Area=6\times 4=24\) Hence Option B is correct. Option A gives the sum, while Options C and D are mathematically unrelated to the rectangle area formula.

❓ Why Other Options Are Incorrect

Option A → Addition of dimensions does not give area.
Option C → 20 is not the product of 6 and 4.
Option D → 2 is unrelated to the dimensions given.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Recognize that area calculations require multiplication of dimensions.

Final Logic:

6 multiplied by 4 gives 24 square units.

🧠 Memory Trick

"Area means multiply sides."

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

📄 Passage

Consider the information regarding the number of men and women workers in three factories I, II and III. Factory I has 30 men and 25 women. Factory II has 25 men and 31 women. Factory III has 27 men and 26 women. This structured information is represented in a 3 x 2 matrix.

Based on the passage, what does the specific entry in the second row and second column represent in the factory matrix?

A Women in Factory I

B Men in Factory III

C Women in Factory II

D Men in Factory II

💡 Short Explanation

Rows represent factories sequentially. Columns represent men and women. Second row-second column means women in Factory II.

✅ Correct Answer Explanation

The matrix rows correspond to Factory I, II, and III respectively, while columns represent men and women. Thus the second row refers to Factory II and the second column refers to women. Therefore the entry represents women workers in Factory II, making Option C correct.

❓ Why Other Options Are Incorrect

Option A → Factory I belongs to the first row, not the second row.
Option B → Men correspond to the first column, not the second column.
Option D → Men in Factory II would be located in the second row-first column.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Match row and column meanings directly from the passage information.

Final Logic:

Second row + second column = women in Factory II.

🧠 Memory Trick

"Rows = factories, Columns = gender"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Tabular Data Representation

📄 Passage

What is the combined numerical total of men workers in Factory I and Factory II?

A 60

B 52

C 56

D 55

💡 Short Explanation

Men in Factory I = 30. Men in Factory II = 25. Total equals 55.

✅ Correct Answer Explanation

The passage states that Factory I has 30 men and Factory II has 25 men. Adding them gives: \(30+25=55\) Thus the combined total is 55, making Option D correct. Other options either exceed or underestimate the correct sum.

❓ Why Other Options Are Incorrect

Option A → 60 exceeds the actual total of men workers.
Option B → 52 results from incorrect arithmetic addition.
Option C → 56 is one more than the correct total.

🎯 Strategy

Used: Substitution

Application:

Extract the relevant values from the passage and add carefully.

Final Logic:

30 + 25 gives exactly 55.

🧠 Memory Trick

"30 plus 25 = 55"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Real-Life Matrix Representation

17 In matrix algebra, what is the exact sum of a 2 x 2 matrix A and a zero matrix of the same mathematical order?

A Zero matrix

B Identity matrix

C Cannot be determined

D Matrix A

💡 Short Explanation

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

✅ Correct Answer Explanation

The zero matrix acts as the additive identity in matrix algebra. Therefore, when a matrix A is added to a zero matrix of the same order, the result remains unchanged: \(A+O=A\) Hence Option D is correct. Other options contradict the identity property of matrix addition.

❓ Why Other Options Are Incorrect

Option A → The result becomes zero only if A itself is a zero matrix.
Option B → Identity matrices relate to multiplication, not addition.
Option C → The result is fully determined by the additive identity property.

🎯 Strategy

Used: Elimination

Application:

Use the additive identity law of matrices to remove incorrect outcomes.

Final Logic:

Adding zero leaves the matrix unchanged.

🧠 Memory Trick

"Zero adds nothing."

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

18 Which statement is INCORRECT regarding the matrix representation of geometry?

A A point (x,y) can be represented as a column matrix.

B A point (x,y) can be represented as a row matrix.

C A quadrilateral's vertices cannot be mathematically combined into a single matrix.

D Matrices assist in accurately representing closed rectilinear figures.

💡 Short Explanation

Geometric coordinates can form matrices. Vertices are easily tabulated together. Hence statement C is incorrect.

✅ Correct Answer Explanation

Matrices are widely used to represent geometric coordinates and shapes. A quadrilateral's vertices can absolutely be organized into a single matrix. Therefore Option C is the incorrect statement. Options A and B are correct because coordinates may be represented as row or column matrices. Option D is also valid.

❓ Why Other Options Are Incorrect

Option A → Coordinates arranged vertically form a valid column matrix.
Option B → Coordinates arranged horizontally form a valid row matrix.
Option D → Matrices help represent geometric figures accurately and compactly.

🎯 Strategy

Used: Odd One Out

Application:

Identify the statement contradicting standard geometric applications of matrices.

Final Logic:

Vertices of geometric figures can indeed be combined into matrices.

🧠 Memory Trick

"Geometry fits into matrices."

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Geometrical Representation

19 Which algebraic properties definitively hold true for matrix addition?
1. Commutative Law.
2. Associative Law.
3. Existence of additive identity.

A 1 only

B 1, 2, and 3

C 2 and 3 only

D 1 and 2 only

💡 Short Explanation

Matrix addition is commutative. Matrix addition is associative. Zero matrix acts as additive identity.

✅ Correct Answer Explanation

Matrix addition satisfies the commutative law, associative law, and existence of additive identity. Specifically: \(A+B=B+A,(A+B)+C=A+(B+C),A+O=A\) Therefore all three statements are true, making Option B correct.

❓ Why Other Options Are Incorrect

Option A → Associative law and additive identity are also valid properties.
Option C → Matrix addition is commutative as well, so Statement 1 cannot be excluded.
Option D → Additive identity is another important valid property of matrix addition.

🎯 Strategy

Used: Option Grouping

Application:

Check each algebraic property individually against matrix addition laws.

Final Logic:

All three standard algebraic properties hold true.

🧠 Memory Trick

"CAA → Commutative, Associative, Additive identity"

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

20 Assertion (A): The negative of matrix A is defined mathematically as -A = (-1)A.
Reason (R): The zero matrix logically acts as the additive identity in matrix algebra.

A Both True, R explains A

B A True, R False

C Both False

D A False, R True

💡 Short Explanation

Negative matrix is scalar multiplication by −1. Zero matrix is additive identity. Together they explain additive inverse property.

✅ Correct Answer Explanation

The negative of a matrix is defined by multiplying every element by −1: \(-A=(-1)A,A+(-A)=O\) The zero matrix acts as the additive identity. Because adding A and −A gives the zero matrix, the Reason correctly explains the Assertion. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → The Reason is true because the zero matrix is indeed the additive identity.
Option C → Both statements are mathematically correct.
Option D → The Assertion is also true according to matrix algebra definitions.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Check whether the Reason logically supports the additive inverse definition.

Final Logic:

Negative matrices exist because their sum with A gives the zero matrix.

🧠 Memory Trick

"Add inverse → Zero matrix"

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

UG Mathematics Booster Test 2 - Introduction to Matrices

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