CUET UG Applied Mathematics Booster Test 2 - Inverse Matrices and Linear Systems -

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CUET UG Applied Mathematics Booster Test 2 - Inverse Matrices and Linear Systems

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 2 - Inverse Matrices and Linear Systems

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QUESTION 1 OF 20

If A and B are square matrices of order n such that AB = BA = I, which of the following is a mathematically strict implication?

Answer Review

1 If A and B are square matrices of order n such that AB = BA = I, which of the following is a mathematically strict implication?

A Matrix A must be a singular matrix.

B |A| must mathematically equal 0.

C Matrix B is precisely defined as A^-1.

D The determinant of B is undefined.

💡 Short Explanation

�� AB = BA = I defines inverse condition �� B acts as inverse of A �� Inverse is unique for square matrices

✅ Correct Answer Explanation

By definition, if AB = BA = I, then B is the inverse of A. → This directly implies B = A⁻¹. → Option A and B contradict invertibility, and D is irrelevant since determinants are defined for square matrices.

❓ Why Other Options Are Incorrect

�� A → Inverse implies A is non-singular, not singular
�� B → If inverse exists, |A| ≠ 0
�� D → Determinant of a square matrix is always defined

🎯 Strategy

Used: Elimination

Application: Use inverse definition to eliminate contradictory statements

Final Logic: Identity product directly defines inverse relationship

🧠 Memory Trick

"AB = I → B = A⁻¹"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Inverse of Matrix

2 Match the specific matrix operations to their theoretical properties and outcomes.

List I	List II
1. Matrix A multiplied by A^-1	a. (I) \|A\| * I
2. A^-1 evaluated when \|A\| = 0	b. B^-1 * A^-1
3. The formula evaluating A(adj A)	c. Identity Matrix
4. Property defining (AB)^-1	d. Strictly does not exist

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

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

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

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

💡 Short Explanation

�� Identity and inverse properties used �� Singular matrices have no inverse �� Adjugate identity relation applied

✅ Correct Answer Explanation

(1) A·A⁻¹ = I → Identity Matrix (c) → (2) |A| = 0 → inverse does not exist (d) → (3) A(adj A) = |A|I → (a) → (4) (AB)⁻¹ = B⁻¹A⁻¹ → (b)

❓ Why Other Options Are Incorrect

�� A → incorrect mapping of adjoint relation
�� B → mismatches identity and inverse rules
�� C → incorrectly swaps inverse and identity mapping

🎯 Strategy

Used: Option Grouping

Application: Match each property with standard theorem

Final Logic: Each identity-based rule uniquely fixes pairing

🧠 Memory Trick

"0 → No inverse, I → Identity, reverse order in inverse product"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Properties of Inverse & Adjoint

3 Assuming A, B, and C are appropriately sized invertible square matrices, which of the following expressions apply valid inverse properties?
1) (ABC)^-1 = C^-1 * B^-1 * A^-1
2) (A^T)^-1 = (A^-1)^T
3) (2A)^-1 = 2 * A^-1
4) The inverse of A is completely unique.

A 1, 3

B 1, 2, 4

C 2, 3, 4

D 1, 2, 3, 4

💡 Short Explanation

�� Product inverse reverses order �� Transpose-inverse property holds �� Uniqueness of inverse is valid

✅ Correct Answer Explanation

1 is correct: (ABC)⁻¹ = C⁻¹B⁻¹A⁻¹ → 2 is correct: (Aᵀ)⁻¹ = (A⁻¹)ᵀ → 4 is correct: inverse is unique → 3 is false since (2A)⁻¹ ≠ 2A⁻¹ (it equals ½A⁻¹)

❓ Why Other Options Are Incorrect

�� 3 → scalar inverse rule is wrong

🎯 Strategy

Used: Elimination

Application: Check standard inverse identities

Final Logic: Scalar rule eliminates option C

🧠 Memory Trick

"Inverse flips order, transpose preserves structure"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Properties of Inverse Matrices

4 Identify the mathematically INCORRECT statement regarding matrix inversion rules:

A True identity property

B |A| can be zero

C Double inverse returns identity matrix

D Non-singular requirement

💡 Short Explanation

�� Double inverse returns original matrix �� Identity is not result of double inverse �� Non-singularity required for inverse

✅ Correct Answer Explanation

(A⁻¹)⁻¹ = A, not identity matrix → So option C is incorrect → Other statements correctly describe inverse properties

❓ Why Other Options Are Incorrect

�� A → correct identity property
�� B → incorrect statement (correct rule is |A| ≠ 0), but option C is more fundamentally wrong inverse law
�� D → correct condition

🎯 Strategy

Used: Extreme Word Filter

Application: Check exact inverse identity law

Final Logic: Double inverse rule directly identifies error

🧠 Memory Trick

"Inverse of inverse = original"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Inverse Properties

5 R₁ ↔ R₃ transformation represents which operation?

A Scalar multiplication

B Equi-multiple addition

C Row interchange transformation

D Column reduction

💡 Short Explanation

�� Rows are swapped �� No scaling involved �� Elementary row operation

✅ Correct Answer Explanation

Swapping two rows is a row interchange operation → It changes determinant sign but not structure type

❓ Why Other Options Are Incorrect

�� A → involves multiplication, not swapping
�� B → involves addition
�� D → column operation, not row

🎯 Strategy

Used: Contextual Matching

Application: Identify operation type from symbol

Final Logic: ↔ indicates interchange

🧠 Memory Trick

"Swap = sign change"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Elementary Row Operations

6 What is the consequence if a row becomes zero after row operations on A = IA?

A Identity matrix

B |A| ≠ 0

C A⁻¹ does not exist

D Transformation error

💡 Short Explanation

�� Zero row implies singular matrix �� Determinant becomes zero �� No inverse exists

✅ Correct Answer Explanation

A zero row implies determinant = 0 → Hence matrix is singular → Singular matrices have no inverse

❓ Why Other Options Are Incorrect

�� A → identity cannot have zero row
�� B → contradicts singularity
�� D → valid operation outcome

🎯 Strategy

Used: Elimination

Application: Use singularity rule

Final Logic: Zero row → determinant zero

🧠 Memory Trick

"Zero row → zero determinant → no inverse"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Invertibility Condition

7 Starting equation for EMI inverse via row operations is:

A A = AI

B I = AA⁻¹

C A = IA

D A = Aᵀ

💡 Short Explanation

�� Identity matrix augmentation used �� Row reduction starts from A = IA �� Leads to inverse formation

✅ Correct Answer Explanation

To find inverse using row operations, we start with A = IA → Then convert A → I to get IA⁻¹

❓ Why Other Options Are Incorrect

�� A → incorrect structure
�� B → true but not starting row operation form
�� D → irrelevant

🎯 Strategy

Used: Substitution

Application: Identify standard setup

Final Logic: Augmented identity form is required

🧠 Memory Trick

"Start with I on right"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Inverse by Row Operations

8 Fundamental goal of C₂ → C₂ − 2C₁ is:

A Swap columns

B Subtract scaled column

C Divide column

D Zero matrix

💡 Short Explanation

�� Elementary column operation �� Linear combination used �� Structure preserved

✅ Correct Answer Explanation

Operation replaces C₂ with C₂ − 2C₁ → Used to simplify matrix without changing determinant rules

❓ Why Other Options Are Incorrect

�� A → swapping not involved
�� C → division not performed
�� D → not a full reduction

🎯 Strategy

Used: Contextual Matching

Application: Interpret operation form

Final Logic: Linear combination defines operation

🧠 Memory Trick

"C₂ minus multiple of C₁"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Column Operations

9 If |A| = 5, value of |adj A| is:

A 5

B 25

C 125

D 1/5

💡 Short Explanation

�� |adj A| = |A|^(n−1) �� For 3×3 matrix, power = 2 �� 5² = 25

✅ Correct Answer Explanation

For n×n matrix, |adj A| = |A|^(n−1) → Here n = 3 → 5² = 25

❓ Why Other Options Are Incorrect

�� A → incorrect scaling
�� C → incorrect exponent
�� D → inverse relation not applicable

🎯 Strategy

Used: Formula Application

Application: Apply adjoint determinant rule

Final Logic: Power rule gives 25

🧠 Memory Trick

"adj raises power (n−1)"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Adjoint Properties

10 Adjoint matrix is formed by:

A Minor matrix

B Identity matrix

C Cofactor matrix

D Scalar matrix

💡 Short Explanation

�� Cofactor matrix is transposed �� Gives adjoint �� Used in inverse formula

✅ Correct Answer Explanation

adj(A) = transpose of cofactor matrix → Hence cofactor matrix is basis of adjoint

❓ Why Other Options Are Incorrect

�� A → minors alone insufficient
�� B → identity unrelated
�� D → scalar matrix irrelevant

🎯 Strategy

Used: Conceptual Matching

Application: Link adjoint definition

Final Logic: Cofactor transpose defines adjoint

🧠 Memory Trick

"Cofactor → transpose → adjoint"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Adjoint Matrix

11 If evaluating the coordinate "area" (determinant) of matrix A yields |A| = -10, and adj A is successfully derived, what exact scalar must be multiplied into adj A to yield A^-1?

A 10

B -1/10

C -10

D 1/10

💡 Short Explanation

�� Inverse uses formula A⁻¹ = (1/|A|) adj A �� Given |A| = -10 �� So scalar becomes -1/10

✅ Correct Answer Explanation

The inverse of a matrix is given by: A^(-1)=1/∣A∣adj⁡(A) → Substituting |A| = -10: A⁻¹ = (1 / -10) adj A = -1/10 × adj A → Hence, the required scalar is -1/10, so Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → 10 ignores division by determinant
�� Option C → -10 is not reciprocal form
�� Option D → 1/10 misses negative sign from determinant

🎯 Strategy

Used: Substitution

Application: Directly substitute determinant value into standard inverse formula

Final Logic: Scalar is always reciprocal of determinant

🧠 Memory Trick

"Inverse = adj over det (flip the sign if det is negative)"

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Adjoint Method of Inverse

12 Using the Adjugate method to find the integral variables of AX = B, if A^-1 = Row 1: (1, 0), Row 2: (0, 1), and B = Column 1: (5, -2), what is the resulting column vector X?

A Column 1: (1, 1)

B Column 1: (5, -2)

C Column 1: (-5, 2)

D Column 1: (10, -4)

💡 Short Explanation

�� X = A⁻¹B �� A⁻¹ is identity matrix �� So X = B

✅ Correct Answer Explanation

From the matrix equation: AX=B → Multiply both sides by A⁻¹: X = A⁻¹B → Given: A⁻¹ = I (identity matrix) → So: X = IB = B → Therefore: X = (5, -2) Therefore, Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → unrelated transformation
�� Option C → incorrect sign change without operation
�� Option D → incorrect scalar multiplication assumption

🎯 Strategy

Used: Substitution

Application: Identity matrix leaves vector unchanged

Final Logic: A⁻¹ = I ⇒ X = B

🧠 Memory Trick

"Identity does nothing."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Inverse and Solution of Linear Equations

13 To solve a 2-variable linear system rigorously utilizing Cramer's Rule, what mathematically occupies the columns of Delta_y?

A The coefficients of y replaced by the coefficients of x.

B The coefficients of y strictly replaced by the constant matrix column.

C The constants entirely replaced by zero.

D The identity matrix components.

💡 Short Explanation

�� Cramer's rule replaces variable column �� Δy replaces y-column with constants �� Determinants are then evaluated

✅ Correct Answer Explanation

In Cramer's Rule: Δ_y is formed by replacing the y-column with constants while keeping all other columns unchanged. → Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → replaces with wrong variable column
�� Option C → unrelated operation
�� Option D → identity matrix is not involved in Cramer's rule

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Identify rule definition of Δy construction

Final Logic: Δy = replace y-column with constants

🧠 Memory Trick

"Y column replaced by constants."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Cramer's Rule

14 If a Cramer's Rule evaluation for variable z yields Delta_z = 32 and Delta = 8, what is the conclusive value of z?

A 40

B 4

C 1/4

D 256

💡 Short Explanation

�� z = Δz / Δ �� Substitute values �� 32 / 8 = 4

✅ Correct Answer Explanation

Using Cramer's Rule: z=Δ_z/Δ → Substitute values: z = 32 / 8 = 4 → Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → incorrect multiplication
�� Option C → inverse ratio error
�� Option D → wrong power operation

🎯 Strategy

Used: Substitution

Application: Direct ratio evaluation

Final Logic: Δz divided by Δ gives variable value

🧠 Memory Trick

"Top over base."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Cramer's Rule Evaluation

15 P1 MCQ In determining system viability, if Delta = 0, Delta_x = 0, Delta_y = 0, and Delta_z = 0, the overall system of equations is classified mathematically as:

A Inconsistent with strictly no solution.

B Dependent consistent, possessing infinitely many solutions.

C Consistent with one strictly unique solution.

D Invalid and unsolvable matrix structures.

💡 Short Explanation

�� All determinants are zero �� System loses uniqueness �� Infinite solutions possible

✅ Correct Answer Explanation

In Cramer's Rule classification: Δ ≠ 0 → unique solution Δ = 0 and at least one Δx ≠ 0 → no solution Δ = Δx = Δy = Δz = 0 → infinitely many solutions → Therefore system is dependent and consistent. → Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → requires at least one non-zero Δx
�� Option C → requires Δ ≠ 0
�� Option D → not a standard classification

🎯 Strategy

Used: Option Grouping

Application: Match determinant conditions with solution types

Final Logic: All zeros ⇒ dependent consistent system

🧠 Memory Trick

"All zero determinants = infinite solutions."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Consistency of Linear Systems

16 If Cramer's rule is applied and Delta evaluates precisely to 0 while Delta_y evaluates to -15, what does this mathematically prove regarding system existence conditions?

A The system is consistent.

B The system has infinitely many solutions.

C The system is inconsistent and possesses absolutely no solution.

D The variable y equals 0.

💡 Short Explanation

�� Δ = 0 �� Δy ≠ 0 �� System is inconsistent

✅ Correct Answer Explanation

In Cramer's Rule: Δ=0, Δ_y≠0 → This condition indicates contradiction in equations. → Therefore, no solution exists. → Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → requires consistency conditions
�� Option B → requires all Δ = 0
�� Option D → variable cannot be directly inferred

🎯 Strategy

Used: Elimination

Application: Match determinant conditions to system behavior

Final Logic: Δ = 0 with Δy ≠ 0 ⇒ no solution

🧠 Memory Trick

"Zero Delta + nonzero Delta → no solution."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Consistency Conditions

17 When formatting a multi-variable system into an augmented matrix for row reduction, where are the values of the constant equations placed geometrically?

A In the primary diagonal.

B As the very first row.

C As the rightmost appended column.

D Scattered throughout the non-diagonal elements.

💡 Short Explanation

�� Augmented matrix = coefficients + constants �� Constants form last column �� Used in row reduction

✅ Correct Answer Explanation

In augmented matrix form: [A∣B] → B (constants) is placed as the rightmost column. → Therefore, Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → diagonal is unrelated
�� Option B → row placement is incorrect
�� Option D → constants are not scattered

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recall structure of augmented matrix

Final Logic: Constants always form rightmost column

🧠 Memory Trick

"Constants sit at the end."

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

18 During the forward elimination stage of row reduction on an augmented matrix, what shape is the coefficient partition of the matrix attempting to achieve?

A Lower triangular matrix

B Scalar matrix

C Upper triangular matrix (leading 1s and trailing zeros below)

D Skew-symmetric matrix

💡 Short Explanation

�� Forward elimination zeros below pivots �� Creates staircase structure �� Leads to back substitution

✅ Correct Answer Explanation

Row reduction transforms the system into an upper triangular form: Upper Triangular Matrix → This structure allows back substitution. → Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → lower triangular is backward elimination
�� Option B → scalar matrix is unrelated
�� Option D → skew-symmetric is unrelated

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match elimination process with matrix shape

Final Logic: Forward elimination produces upper triangular form

🧠 Memory Trick

"Forward = Upper staircase."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Row Reduction Method

📄 Passage

When assessing the viability of a linear system using determinant techniques, the Coefficient determinant (Delta) holds ultimate authority. If Delta operates strictly as a non-zero integer or rational, the system safely guarantees a unique operational solution regardless of the constant matrix outcomes. However, if Delta crashes to zero, the system fractures into two possibilities: if any corresponding variable determinants (Delta_x, Delta_y) contain non-zero values, parallel mathematical lines manifest, meaning the system is entirely inconsistent with no crossing solutions.

Question:
Based strictly on the passage, what is the ultimate mathematical authority for determining if a linear system possesses a unique solution?

A The Identity Matrix

B The Coefficient determinant (Delta)

C Delta_x exclusively

D The presence of parallel lines

💡 Short Explanation

�� Determinant controls uniqueness �� Δ ≠ 0 → unique solution �� Δ is governing factor

✅ Correct Answer Explanation

The passage states that Δ determines system behavior. → If Δ ≠ 0, unique solution exists. → Therefore Δ is the controlling quantity. → Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → identity matrix is not decision criterion
�� Option C → Δx is not primary authority
�� Option D → geometric interpretation, not algebraic control

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Identify controlling variable in passage

Final Logic: Δ governs uniqueness

🧠 Memory Trick

"Delta decides destiny."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Consistency of Linear Systems

📄 Passage

Question:
If Delta crashes to zero and Delta_x evaluates to 12, what geometric and algebraic phenomenon occurs according to the passage?

A The system guarantees a unique solution of x = 12.

B Parallel mathematical lines manifest, meaning the system is inconsistent with no solutions.

C The lines intersect perfectly at the origin.

D The system resolves to an infinite multitude of solutions.

💡 Short Explanation

�� Δ = 0 breaks uniqueness �� Δx ≠ 0 indicates inconsistency �� Parallel lines imply no solution

✅ Correct Answer Explanation

From the passage: Δ=0, Δ_x≠0 → This implies inconsistent system. → Geometrically, lines are parallel and do not intersect. → Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → requires Δ ≠ 0
�� Option C → would require consistent intersection
�� Option D → occurs only when all Δ = 0

🎯 Strategy

Used: Elimination

Application: Match determinant conditions to geometric interpretation

Final Logic: Δ = 0 and Δx ≠ 0 ⇒ no solution (parallel lines)

🧠 Memory Trick

"Zero Delta + nonzero Delta = parallel lines."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Consistency of Linear Systems

CUET UG Applied Mathematics Booster Test 2 – Inverse Matrices and Linear Systems

CUET UG Applied Mathematics Booster Test 2 - Inverse Matrices and Linear Systems

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