UG Mathematics Booster Test 3 - Introduction to Matrices -

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UG Mathematics Booster Test 3 - Introduction to Matrices

⏱️ 30 Minutes

❓ 20 Questions

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

UG Mathematics Booster Test 3 - Introduction to Matrices

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QUESTION 1 OF 20

What mathematically defines a square matrix array?

Answer Review

1 What mathematically defines a square matrix array?

A m > n

B m = n

C m < n

D It contains only real numbers

💡 Short Explanation

Square matrices have equal rows and columns. Order is written as n × n. Hence m equals n.

✅ Correct Answer Explanation

A square matrix is defined as a matrix having the same number of rows and columns. Therefore its order satisfies m = n. Option B correctly states this condition. Options A and C describe unequal dimensions, while Option D relates to matrix entries rather than the defining property of square matrices.

❓ Why Other Options Are Incorrect

Option A → m > n represents matrices with more rows than columns, not square matrices.
Option C → m < n represents matrices with more columns than rows.
Option D → Square matrices may contain real or complex numbers; entry type does not define squareness.

🎯 Strategy

Used: Elimination

Application:

Check which condition matches the standard definition of a square matrix.

Final Logic:

Equal rows and columns imply m = n.

🧠 Memory Trick

"Square → Same rows and columns"

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

2 Which statement is analytically INCORRECT regarding matrix utility?

A Matrices cannot be used in sales projection and cost estimation.

B Matrices are heavily used in genetics.

C Matrices are used in industrial management.

D Matrices are used in economics and sociology.

💡 Short Explanation

Matrices support business calculations. They are useful in management and economics. Option A falsely denies applications.

✅ Correct Answer Explanation

Matrices are widely used in sales projection, budgeting, cost estimation, economics, genetics, and industrial management. Therefore Option A is incorrect because matrices are extremely useful in such business applications. Options B, C, and D correctly describe recognized real-life uses of matrices in data handling and analysis.

❓ Why Other Options Are Incorrect

Option B → Genetics uses matrices for population studies and data organization.
Option C → Industrial management applies matrices in optimization and resource allocation.
Option D → Economics and sociology frequently use matrix-based statistical models.

🎯 Strategy

Used: Extreme Word Filter

Application:

Identify overly absolute claims such as "cannot be used."

Final Logic:

Matrices are extensively used in sales and cost estimation.

🧠 Memory Trick

"Matrices manage business data."

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

3 Assertion (A): Matrices cannot be used to solve systems of linear equations.
Reason (R): Matrices strictly represent geometric reflections only and have no algebraic utility.

A Both True, R explains A

B A True, R False

C A False, R True

D Both False

💡 Short Explanation

Matrices solve linear equations efficiently. They have strong algebraic applications. They are not limited to reflections only.

✅ Correct Answer Explanation

Matrices are extensively used for solving systems of linear equations through methods such as inverse matrices and row operations. Therefore the Assertion is false. The Reason is also false because matrices possess wide algebraic utility beyond geometry. Hence both statements are incorrect, making Option D the correct answer.

❓ Why Other Options Are Incorrect

Option A → Both statements are mathematically false.
Option B → The Assertion itself is incorrect because matrices solve linear systems.
Option C → The Reason is also false since matrices are not restricted to geometric reflections.

🎯 Strategy

Used: Elimination

Application:

Verify Assertion and Reason separately using standard matrix applications.

Final Logic:

Matrices are both algebraically useful and equation-solving tools.

🧠 Memory Trick

"Matrices solve equations too."

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

4 Arrange the analytical steps for successful matrix multiplication C = AB:
1. Ensure columns of A equals rows of B.
2. Take ith row of A.
3. Take kth column of B.
4. Multiply elements and sum them.

A 1-3-2-4

B 4-3-2-1

C 1-2-3-4

D 2-1-4-3

💡 Short Explanation

Multiplication requires compatible orders. Rows and columns are selected systematically. Products are summed to form entries.

✅ Correct Answer Explanation

For matrix multiplication, first verify compatibility: columns of A must equal rows of B. Then select the ith row of A and kth column of B. Multiply corresponding elements and add them to obtain the matrix entry. Hence the correct sequence is 1-2-3-4, making Option C correct.

❓ Why Other Options Are Incorrect

Option A → Column selection occurs after choosing the row of A.
Option B → Multiplication steps cannot occur before compatibility checking.
Option D → Compatibility must be verified before computation begins.

🎯 Strategy

Used: Option Grouping

Application:

Follow the logical sequence of matrix multiplication operations.

Final Logic:

Compatibility check always comes before multiplication.

🧠 Memory Trick

"Check → Row → Column → Multiply"

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

5 A company uses a structured matrix to formally track 3 products across 5 regions. What is the total numerical number of entries?

A 15

B 8

C 10

D 20

💡 Short Explanation

Total entries = rows × columns. Matrix dimensions are 3 × 5. Product equals 15.

✅ Correct Answer Explanation

The number of entries in a matrix equals the product of rows and columns: \(3\times 5=15\) Therefore the matrix contains 15 entries, making Option A correct. Other options do not satisfy the formula for total matrix elements.

❓ Why Other Options Are Incorrect

Option B → 8 is much smaller than the required number of entries.
Option C → 10 corresponds to incorrect multiplication.
Option D → 20 would require a 4×5 matrix.

🎯 Strategy

Used: Substitution

Application:

Multiply rows and columns directly to find total entries.

Final Logic:

3 × 5 gives exactly 15.

🧠 Memory Trick

"Entries = Rows × Columns"

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

6 A 3 x 3 matrix has numerical entries that are exclusively either 1 or 2. If one matrix is randomly constructed, what is the exact probability that all 9 elements are identically 1?

A 1/9

B 1/81

C 1/512

D 1/256

💡 Short Explanation

Each entry has two possibilities. Total matrices = 2⁹. Only one matrix has all entries 1.

✅ Correct Answer Explanation

A 3×3 matrix contains 9 entries. Each entry may independently be either 1 or 2, giving total possible matrices: \(2^{9}=512\) Only one matrix has all entries equal to 1. Hence probability = 1/512, making Option C correct.

❓ Why Other Options Are Incorrect

Option A → 1/9 ignores the independent possibilities of all entries.
Option B → 1/81 corresponds incorrectly to powers of 3 rather than 2.
Option D → 1/256 would arise from only 8 independent entries.

🎯 Strategy

Used: Substitution

Application:

Count total possible matrices using independent choices per entry.

Final Logic:

One favorable outcome among 512 total outcomes gives 1/512.

🧠 Memory Trick

"2 choices repeated 9 times → 2⁹"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Matrix Formation and Counting

7 A mathematical geometric 3D vector [x, y, z] is represented strictly as a column matrix. What is its formal order?

A 2 x 1

B 1 x 3

C 3 x 3

D 3 x 1

💡 Short Explanation

Column matrices arrange entries vertically. Three coordinates form three rows. One column remains throughout.

✅ Correct Answer Explanation

A 3D vector represented as a column matrix appears vertically: \(\left[\begin{aligned}x\\ y\\ z\end{aligned}\right]\) This matrix has 3 rows and 1 column, so its order is 3×1. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → A 2×1 matrix contains only two entries.
Option B → 1×3 represents a row matrix, not a column matrix.
Option C → A 3×3 matrix requires nine entries.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Count vertical entries and determine the column structure.

Final Logic:

Three vertically stacked coordinates form a 3×1 matrix.

🧠 Memory Trick

"3D column → 3×1"

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

8 A graph region reflection uses a 2 x 2 identity matrix multiplied by the scalar numerical value -1. What is the resulting matrix array?

A [1 0; 0 1]

B [-1 0; 0 -1]

C [0 -1; -1 0]

D [-1 -1; -1 -1]

💡 Short Explanation

Identity matrix has diagonal ones. Multiplying by −1 changes signs. Non-diagonal zeros remain unchanged.

✅ Correct Answer Explanation

The identity matrix is: \(I=\left[\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\right]\) Multiplying by −1 gives: \(-I=\left[\begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix}\right]\) Thus Option B is correct. The other matrices do not result from scalar multiplication of the identity matrix.

❓ Why Other Options Are Incorrect

Option A → This is the original identity matrix before multiplication by −1.
Option C → Off-diagonal values incorrectly become −1.
Option D → Scalar multiplication does not change zero entries into −1.

🎯 Strategy

Used: Substitution

Application:

Multiply every entry of the identity matrix by −1 systematically.

Final Logic:

Diagonal ones become −1 while zeros remain unchanged.

🧠 Memory Trick

"−I flips only diagonal signs."

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

9 Match the specific tabular matrix types:

List I	List II
1. Null matrix	a. All elements 0
2. Null matrix	b. Non-diagonal 0
3. Identity matrix	c. Diagonal 1, rest 0

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

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

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

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

💡 Short Explanation

Null matrix contains all zeros. Diagonal matrix has off-diagonal zeros. Identity matrix has diagonal ones.

✅ Correct Answer Explanation

A null matrix contains only zeros. A diagonal matrix has all non-diagonal entries equal to zero. An identity matrix has diagonal entries equal to 1 with all remaining entries zero. Hence the correct matching is 1-a, 2-b, 3-c, making Option C correct.

❓ Why Other Options Are Incorrect

Option A → Diagonal matrices do not require diagonal entries to be all ones.
Option B → Identity and null matrix properties are interchanged incorrectly.
Option D → Null matrices cannot merely have non-diagonal zeros; all entries must be zero.

🎯 Strategy

Used: Option Grouping

Application:

Match standard matrix definitions with their exact structural properties.

Final Logic:

Only Option C correctly pairs all matrix types.

🧠 Memory Trick

"Null all 0, Identity diagonal 1"

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

10 Which of the following algebraically describe a scalar matrix?
1. It is a diagonal matrix.
2. Its diagonal elements are equal.
3. It is a square matrix.

A 1 and 2 only

B 1, 2, and 3

C 1 and 3 only

D 2 and 3 only

💡 Short Explanation

Scalar matrices are diagonal matrices. Diagonal entries are equal. Scalar matrices are always square.

✅ Correct Answer Explanation

A scalar matrix is a square diagonal matrix in which all diagonal entries are equal. Therefore all three statements are correct. Option B correctly describes the properties of scalar matrices. Any omission of these conditions gives an incomplete definition.

❓ Why Other Options Are Incorrect

Option A → Scalar matrices must also be square matrices.
Option C → Equal diagonal elements are essential and cannot be ignored.
Option D → A scalar matrix must specifically be diagonal as well.

🎯 Strategy

Used: Option Grouping

Application:

Check each defining property of scalar matrices individually.

Final Logic:

All three statements together define scalar matrices completely.

🧠 Memory Trick

"Scalar = Square + Diagonal + Equal diagonals"

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

11 The generic matrix element denoted inside brackets as a_ij algebraically lies in which specific row and column?

A ith row, jth column

B jth row, ith column

C ith row, ith column

D jth row, jth column

💡 Short Explanation

Subscript i represents row number. Subscript j represents column number. aᵢⱼ identifies exact matrix position.

✅ Correct Answer Explanation

In matrix notation, aᵢⱼ denotes the element located in the ith row and jth column. This is the standard convention used in NCERT mathematics. Therefore Option A is correct. The remaining options incorrectly interchange or duplicate row-column positions.

❓ Why Other Options Are Incorrect

Option B → Row and column indices are reversed incorrectly.
Option C → Both subscripts cannot simultaneously represent the row position.
Option D → Both subscripts cannot simultaneously represent the column position.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Interpret the standard meaning of matrix subscripts carefully.

Final Logic:

First index = row, second index = column.

🧠 Memory Trick

"i → row, j → column"

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

12 A 1×1 matrix element a_11 is determined by the numerical integral
∫₀² 4x³ dx
Find a_11.

A 8

B 32

C 64

D 16

💡 Short Explanation

Integrate 4x³ carefully. Antiderivative becomes x⁴. Applying limits gives 16.

✅ Correct Answer Explanation

Evaluate the definite integral: \(\int_{0}^{2}\,4x^{3} dx={\left[x^{4}\right]}_{0}^{2}=16\) Thus the matrix element a₁₁ equals 16. Therefore Option D is correct. Other options arise from incorrect integration or substitution of limits.

❓ Why Other Options Are Incorrect

Option A → 8 results from incomplete power evaluation.
Option B → Incorrect arithmetic leads to this value.
Option C → 64 equals 2⁶, not the evaluated integral.

🎯 Strategy

Used: Substitution

Application:

Integrate first and then apply upper and lower limits correctly.

Final Logic:

2⁴ − 0⁴ gives 16.

🧠 Memory Trick

"4x³ integrates to x⁴"

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

📄 Passage

A political group hired a firm to promote its candidate in three ways: telephone, house calls, and letters. The cost per contact is given by column matrix A = [40; 100; 50]. The number of contacts in two cities X and Y are given by matrix B.

In the given column matrix A = [40; 100; 50] from the passage, how many horizontal rows are there?

A 1

B 2

C 4

D 3

💡 Short Explanation

Column matrices arrange entries vertically. Three entries are listed. Hence there are three rows.

✅ Correct Answer Explanation

The column matrix A contains three vertically arranged entries: 40, 100, and 50. \(A=\left[\begin{aligned}40\\ 100\\ 50\end{aligned}\right]\) Therefore the matrix has 3 rows and 1 column. Hence Option D is correct. Other options incorrectly count the number of entries.

❓ Why Other Options Are Incorrect

Option A → One row would correspond to a row matrix, not a column matrix.
Option B → Only two entries would produce two rows.
Option C → The matrix contains only three vertically arranged entries.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Count the number of vertically arranged entries in the column matrix.

Final Logic:

Three entries vertically imply three rows.

🧠 Memory Trick

"Vertical entries = row count"

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

📄 Passage

If matrix A from the passage is logically multiplied by a scalar value of 2 to represent doubled numerical costs, what is the new matrix cost for house calls?

A 100

B 80

C 200

D 150

💡 Short Explanation

House call cost equals 100. Scalar multiplication doubles entries. New cost becomes 200.

✅ Correct Answer Explanation

The house-call entry in matrix A equals 100. Multiplying the entire matrix by scalar 2 doubles every entry: \(2\times 100=200\) Therefore the new house-call cost becomes 200, making Option C correct.

❓ Why Other Options Are Incorrect

Option A → 100 is the original cost before scalar multiplication.
Option B → 80 corresponds incorrectly to doubling the telephone cost.
Option D → 150 does not result from multiplying 100 by 2.

🎯 Strategy

Used: Substitution

Application:

Identify the relevant matrix entry and multiply by the scalar value.

Final Logic:

Doubling 100 gives 200.

🧠 Memory Trick

"Scalar multiply → every entry changes"

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

15 A rectangular mathematical field has physical dimensions given by a 2 x 1 matrix [10; 5]. What is the calculated area of this field in square units?

A 15

B 50

C 20

D 2

💡 Short Explanation

Rectangle area = length × breadth. Dimensions are 10 and 5. Product equals 50.

✅ Correct Answer Explanation

The dimensions represented in the matrix are 10 and 5. The rectangular area is calculated as: \(10\times 5=50\) Hence the area equals 50 square units, making Option B correct. Other options result from incorrect arithmetic operations.

❓ Why Other Options Are Incorrect

Option A → 15 comes from addition instead of multiplication.
Option C → 20 does not equal the product of the dimensions.
Option D → 2 is unrelated to the given measurements.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Recognize that area requires multiplication of dimensions.

Final Logic:

10 multiplied by 5 gives 50.

🧠 Memory Trick

"Area means multiply sides."

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

16 The numerical test scores for 4 persons are given in a matrix array as 50, 60, 70, 80. What is the analytical moving average (mean) of their scores?

A 65

B 60

C 70

D 75

💡 Short Explanation

Mean = Total ÷ Number of values. Total score equals 260. Average equals 65.

✅ Correct Answer Explanation

The mean of the scores is calculated by adding all values and dividing by the total number of observations: \(\frac{50+60+70+80}{4}=65\) Thus the average score is 65, making Option A correct.

❓ Why Other Options Are Incorrect

Option B → 60 is below the correct arithmetic mean.
Option C → 70 is larger than the actual average.
Option D → 75 results from incorrect averaging.

🎯 Strategy

Used: Substitution

Application:

Insert the values into the arithmetic mean formula directly.

Final Logic:

260 divided by 4 equals 65.

🧠 Memory Trick

"Mean = Sum ÷ Count"

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

17 Assertion (A): For transpose matrices, (A′)′ = A.
Reason (R): Transposing a matrix twice restores its original row and column arrangement.

A Both False

B Both True, R explains A

C A True, R False

D A False, R True

💡 Short Explanation

First transpose interchanges rows and columns. Second transpose restores original arrangement. Hence matrix becomes unchanged again.

✅ Correct Answer Explanation

The transpose operation interchanges rows and columns. Applying transpose again restores the original arrangement: \(\left(A^{T})^{T},\ A\right.\) Therefore both the Assertion and Reason are true, and the Reason correctly explains the Assertion. Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Both statements are standard transpose properties and are true.
Option C → The Reason is also true and logically explains the Assertion.
Option D → The Assertion itself is correct according to transpose rules.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Verify both statements and check explanatory relationship.

Final Logic:

Double transposition restores the original matrix.

🧠 Memory Trick

"Transpose twice = original matrix"

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

18 Which statement about symmetric matrices is analytically incorrect?

A A symmetric matrix can be a column matrix of order 3×1

B A matrix is symmetric if A′ = A

C Elements satisfy a_ij = a_ji

D A symmetric matrix must be square

💡 Short Explanation

Symmetric matrices must be square. Column matrices are generally non-square. Hence Option A is incorrect.

✅ Correct Answer Explanation

A symmetric matrix satisfies: \(A^{T}=A\) Such matrices must necessarily be square because transpose equality requires equal rows and columns. Therefore a 3×1 column matrix cannot be symmetric. Hence Option A is the incorrect statement.

❓ Why Other Options Are Incorrect

Option B → This is the standard definition of a symmetric matrix.
Option C → Symmetric matrices satisfy aᵢⱼ = aⱼᵢ for all entries.
Option D → Every symmetric matrix must indeed be square.

🎯 Strategy

Used: Odd One Out

Application:

Identify the statement violating the definition of symmetric matrices.

Final Logic:

Non-square matrices cannot satisfy symmetry conditions.

🧠 Memory Trick

"Symmetric ⇒ Square"

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

19 Which of the following are valid matrix operations?
1. Addition of matrices of the same order
2. Multiplication of a matrix by a scalar
3. Addition of matrices of different orders

A 2 and 3 only

B 1 and 3 only

C 1, 2, and 3

D 1 and 2 only

💡 Short Explanation

Same-order matrices can be added. Scalar multiplication is always valid. Different-order matrices cannot be added.

✅ Correct Answer Explanation

Matrix addition is defined only for matrices of the same order. Scalar multiplication is always permitted. However, matrices of different orders cannot be added. Therefore Statements 1 and 2 are correct while Statement 3 is false, making Option D correct.

❓ Why Other Options Are Incorrect

Option A → Addition of different-order matrices is invalid.
Option B → Scalar multiplication is also a valid operation and cannot be excluded.
Option C → Statement 3 is mathematically incorrect.

🎯 Strategy

Used: Elimination

Application:

Check validity conditions for each matrix operation separately.

Final Logic:

Only Statements 1 and 2 satisfy matrix operation rules.

🧠 Memory Trick

"Same order for addition."

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

20 Match the algebraic matrix rule to its formal term:

List I	List II
1. AB = BA = I	a. Skew-symmetric matrix
2. A′ = A	b. Invertible matrix (inverse)
3. A′ = −A	c. Symmetric matrix

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

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

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

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

💡 Short Explanation

AB = BA = I defines inverse matrices. A′ = A defines symmetric matrices. A′ = −A defines skew-symmetric matrices.

✅ Correct Answer Explanation

If AB = BA = I, then B is the inverse of A. A matrix satisfying A′ = A is symmetric, while A′ = −A defines a skew-symmetric matrix. Therefore the correct matching is 1-b, 2-c, 3-a, making Option C correct.

❓ Why Other Options Are Incorrect

Option A → Symmetric and skew-symmetric definitions are interchanged incorrectly.
Option B → Invertible matrices are incorrectly matched with skew symmetry.
Option D → A′ = A cannot define invertibility.

🎯 Strategy

Used: Option Grouping

Application:

Match each matrix equation with its standard algebraic definition.

Final Logic:

Only Option C correctly pairs all three matrix properties.

🧠 Memory Trick

"Transpose same → symmetric, negative → skew"

📚 Class XI – NCERT Mathematics – Chapter: Matrices – Topic: Symmetric and Inverse Matrices

UG Mathematics Booster Test 3 - Introduction to Matrices

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