CUET UG Mathematics Booster Test 2 - Adjoint, Inverse Matrices and Applications of Determinants -

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CUET UG Mathematics Booster Test 2 - Adjoint, Inverse Matrices and Applications of Determinants

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Mathematics Booster Test 2 - Adjoint, Inverse Matrices and Applications of Determinants

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QUESTION 1 OF 20

Which statements are true for a square matrix \(A\)?
I. \(adj⁡(A)\)is the transpose of the cofactor matrix
II. \(A(adj⁡A)=∣A∣I\)
III. \(adj⁡(A)\)is always symmetric

Answer Review

1 Which statements are true for a square matrix \(A\)?
I. \(adj⁡(A)\)is the transpose of the cofactor matrix
II. \(A(adj⁡A)=∣A∣I\)
III. \(adj⁡(A)\)is always symmetric

A I only

B II only

C I and II

D I, II, and III

💡 Short Explanation

Adjoint is transpose of cofactor matrix. Matrix-adjoint product equals determinant times identity. Adjoint need not always be symmetric.

✅ Correct Answer Explanation

The adjoint of a matrix is defined as the transpose of its cofactor matrix. Also: \(A(adj⁡A)=∣A∣I\) Thus statements I and II are true. However, \(adj⁡(A)\)is not always symmetric unless the original matrix has special properties. Therefore statement III is false. Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores the standard identity \(A(adj⁡A)=∣A∣I\).
Option B → Omits the definition of adjoint as transpose of cofactor matrix.
Option D → Adjoint matrices are not necessarily symmetric for arbitrary matrices.

🎯 Strategy

Used: Elimination

Application:

Check each statement against standard adjoint matrix properties.

Final Logic:

Only statements I and II hold universally.

🧠 Memory Trick

"Adjoint = transpose of cofactors."

📚 Class XII – Mathematics Part 1 – Chapter: Determinants – Topic: Adjoint of a Matrix

2 Arrange the steps to form \(adj⁡(A)\):
1. Identify element \(a_{ij}\)
2. Compute minor \(M_{ij}\)
3. Form cofactor \(A_{ij}=(-1)^{\left(i,\ j\right)}M_{ij}\)
4. Transpose the cofactor matrix

A 1, 2, 3, 4

B 2, 3, 4, 1

C 1, 3, 2, 4

D 4, 3, 2, 1

💡 Short Explanation

Begin with matrix elements. Find minors and cofactors systematically. Transpose final cofactor matrix.

✅ Correct Answer Explanation

To form \(adj⁡(A)\): 1. Identify matrix elements. 2. Compute corresponding minors. Apply sign rule to obtain cofactors: 3. \(A_{ij}=(-1)^{i+j}M_{ij}\) 4. Transpose the cofactor matrix. Thus the correct sequence is 1,2,3,4. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Cannot compute cofactors before identifying matrix positions properly.
Option C → Cofactors require minors first; order is reversed incorrectly.
Option D → Transpose is the final step, not the first.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Follow the logical construction process of adjoint matrices step-by-step.

Final Logic:

Minor precedes cofactor, transpose comes last.

🧠 Memory Trick

"Minor → cofactor → transpose."

📚 Class XII – Mathematics Part 1 – Chapter: Determinants – Topic: Construction of Adjoint

3 Match expressions:

List I	List II
1. \(A(adj⁡A)\)	a. \(A_{ji}\)
2. \(∣adj⁡A∣\)	b. \(∣A∣^{2}\)(for 3×3)
3. Elements of \(\operatorname{adj}\,A\)	c. \(∣A∣I\)

A 1–a, 2–b, 3–c

B 1–b, 2–c, 3–a

C 1–a, 2–c, 3–b

D 1–c, 2–b, 3–a

💡 Short Explanation

Product with adjoint gives determinant identity. Adjoint determinant follows exponent rule. Adjoint entries are transposed cofactors.

✅ Correct Answer Explanation

For a square matrix: \(A(adj⁡A)=∣A∣I\) Thus 1–c. For a 3×3 matrix: \(∣adj⁡A∣=∣A∣^{2}\) giving 2–b. Elements of adjoint are cofactors arranged transposed, hence \(A_{ji}\), giving 3–a. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Confuses determinant identity with adjoint entries.
Option B → Product identity and determinant rule are mismatched.
Option C → Incorrectly swaps determinant property and adjoint elements.

🎯 Strategy

Used: Option Grouping

Application:

Associate each standard adjoint formula with its correct interpretation.

Final Logic:

Adjoint identities uniquely determine the matching order.

🧠 Memory Trick

"Adjoint flips cofactors."

📚 Class XII – Mathematics Part 1 – Chapter: Determinants – Topic: Properties of Adjoint

4 If \(∣A∣=4\) for a 3×3 matrix, find \(∣adj⁡A∣\).
\(∣adj⁡A∣=∣A∣^{\left(n,\ 1\right)}=4^{2}=16\)

A 4

B 16

C 64

D 8

💡 Short Explanation

Use adjoint determinant property. For order 3, exponent becomes 2. \(4^{2}=16\).

✅ Correct Answer Explanation

For an \(n\times n\) matrix: \(∣adj⁡A∣=∣A∣^{n-1}\) Since matrix order is 3: \(∣adj⁡A∣=4^{2}=16\) Hence Option B is correct. Other options use incorrect exponents or direct determinant copying.

❓ Why Other Options Are Incorrect

Option A → Uses determinant directly instead of squaring.
Option C → Incorrectly cubes the determinant.
Option D → Does not satisfy adjoint determinant formula.

🎯 Strategy

Used: Substitution

Application:

Apply adjoint-determinant relation carefully using matrix order.

Final Logic:

For 3×3 matrices, exponent equals 2.

🧠 Memory Trick

"Adjoint power = one less than order."

📚 Class XII – Mathematics Part 1 – Chapter: Determinants – Topic: Determinant of Adjoint

5 Assertion (A): If \(A\) is singular, \(A^{-1}\)does not exist.
Reason (R):
\(A^{-1}=\frac{\operatorname{adj}\,A}{∣A∣}\)
So, division by zero is not defined.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Singular matrices have zero determinant. Inverse formula divides by determinant. Division by zero is impossible.

✅ Correct Answer Explanation

A matrix is singular when: \(∣A∣=0\) The inverse formula is: \(A^{-1}=\frac{\operatorname{adj}\,A}{∣A∣}\) Since division by zero is undefined, inverse cannot exist for singular matrices. Thus both Assertion and Reason are true, and Reason correctly explains Assertion.

❓ Why Other Options Are Incorrect

Option A → Both statements are valid determinant properties.
Option B → Reason is mathematically correct and explains the assertion.
Option D → Assertion is true because singular matrices are noninvertible.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Connect inverse existence directly with determinant value.

Final Logic:

Zero determinant prevents inverse formation.

🧠 Memory Trick

"Zero determinant → no inverse."

📚 Class XII – Mathematics Part 1 – Chapter: Determinants – Topic: Inverse of Matrix

6 If a matrix maps non-zero vectors to the zero vector, then:

A Identity

B Invertible

C Non-invertible

D Orthogonal

💡 Short Explanation

Invertible matrices preserve nonzero vectors. Mapping to zero implies information loss. Such matrices are singular.

✅ Correct Answer Explanation

If a nonzero vector is mapped to the zero vector, the transformation loses uniqueness. Hence the matrix cannot have an inverse. Such matrices are singular or non-invertible. Therefore Option C is correct. Identity and orthogonal matrices always preserve nonzero vectors.

❓ Why Other Options Are Incorrect

Option A → Identity matrix maps vectors to themselves, never to zero.
Option B → Invertible matrices cannot collapse nonzero vectors into zero.
Option D → Orthogonal matrices preserve vector lengths and directions.

🎯 Strategy

Used: Elimination

Application:

Interpret matrix invertibility geometrically through vector mapping.

Final Logic:

Collapsing vectors to zero destroys invertibility.

🧠 Memory Trick

"Zero output means no inverse."

📚 Class XII – Mathematics Part 1 – Chapter: Matrices – Topic: Invertible Transformations

7 If a determinant is non-zero, then the matrix is:

A Singular

B Null

C Non-singular

D Zero matrix

💡 Short Explanation

Nonzero determinant implies invertibility. Such matrices are nonsingular. Singular matrices have determinant zero.

✅ Correct Answer Explanation

A square matrix with: \(∣A∣\neq 0\) is called nonsingular and possesses an inverse. Hence Option C is correct. Singular matrices specifically have determinant zero. Null or zero matrices usually have determinant zero and are noninvertible.

❓ Why Other Options Are Incorrect

Option A → Singular matrices always have determinant zero.
Option B → Null matrices cannot have nonzero determinant.
Option D → Zero matrices have determinant zero, not nonzero.

🎯 Strategy

Used: Elimination

Application:

Apply the determinant criterion for matrix classification.

Final Logic:

Nonzero determinant defines nonsingular matrices.

🧠 Memory Trick

"Nonzero determinant = nonsingular."

📚 Class XII – Mathematics Part 1 – Chapter: Determinants – Topic: Singular and Nonsingular Matrices

8 Identify incorrect statement:

A An invertible matrix can have determinant =0

B If invertible, \(AB=BA=I\) for some \(B\)

C Inverse is unique

D \(\left(A^{-1})^{-1},\ A\right.\)

💡 Short Explanation

Invertible matrices require nonzero determinant. Matrix inverses are unique. Double inverse returns original matrix.

✅ Correct Answer Explanation

An invertible matrix must satisfy: \(∣A∣\neq 0\) Hence Option A is incorrect. Option B correctly defines inverse matrices. Option C is true because inverses are unique. Option D is also correct: \(\left(A^{-1})^{-1},\ A\right.\)

❓ Why Other Options Are Incorrect

Option B → Correct inverse definition using identity matrix multiplication.
Option C → A matrix cannot possess two distinct inverses.
Option D → Taking inverse twice restores the original matrix.

🎯 Strategy

Used: Extreme Word Filter

Application:

Check whether determinant-zero condition contradicts invertibility.

Final Logic:

Invertibility always requires nonzero determinant.

🧠 Memory Trick

"No inverse when determinant vanishes."

📚 Class XII – Mathematics Part 1 – Chapter: Matrices – Topic: Invertible Matrices

9 \(∣A∣=\int_{1}^{e}\,\frac{1}{x} dx=ln⁡e=1adj⁡A=\left(\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}\right)A^{-1}=\frac{1}{1}\left(\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}\right)=\left(\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}\right)\)

A \(\left(\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\right)\)

B \(\left(\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}\right)\)

C \(\left(\begin{pmatrix}0.5 & 0\\ 0 & 0.5\end{pmatrix}\right)\)

D \(\left(\begin{pmatrix}e & 0\\ 0 & e\end{pmatrix}\right)\)

💡 Short Explanation

Determinant evaluates to 1. Inverse formula uses adjoint divided by determinant. Matrix remains unchanged after division.

✅ Correct Answer Explanation

Since: \(∣A∣=\int_{1}^{e}\,\frac{1}{x}dx=1\) the inverse formula becomes: \(A^{-1}=\frac{\operatorname{adj}\,A}{∣A∣}=\frac{1}{1}\left(\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}\right)\) Thus: \(A^{-1}=\left(\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}\right)\) Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Identity matrix would arise only if adjoint itself were identity.
Option C → Incorrect reciprocal scaling has been applied.
Option D → Determinant value equals 1, not \(e\).

🎯 Strategy

Used: Substitution

Application:

Evaluate determinant first, then apply inverse formula directly.

Final Logic:

Division by 1 leaves adjoint unchanged.

🧠 Memory Trick

"Determinant one keeps adjoint same."

📚 Class XII – Mathematics Part 1 – Chapter: Determinants – Topic: Inverse Formula

10 For a matrix transformation to be reversible:

A \(∣A∣=0\)

B Area becomes zero

C Adjoint is zero

D \(∣A∣\neq 0\)

💡 Short Explanation

Reversible transformations require invertibility. Invertibility requires nonzero determinant. Zero determinant collapses dimensions.

✅ Correct Answer Explanation

A matrix transformation is reversible only when the matrix has an inverse. This occurs precisely when: \(∣A∣\neq 0\) Zero determinant implies loss of dimension or area collapse, making reversal impossible. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Determinant zero produces singular, nonreversible transformations.
Option B → Zero area indicates collapse and loss of invertibility.
Option C → Adjoint may be nonzero even for singular matrices.

🎯 Strategy

Used: Elimination

Application:

Connect geometric reversibility with determinant condition.

Final Logic:

Only nonzero determinants permit inverse transformations.

🧠 Memory Trick

"Reversible means determinant nonzero."

📚 Class XII – Mathematics Part 1 – Chapter: Determinants – Topic: Invertibility Condition

11 Determinants \(∣A∣\)and \(∣B∣\)are independently chosen from \(\left\{1,\ 2\right\}\).
Find \(P(∣AB∣=2)\), given \(∣AB∣=∣A∣∣B∣\).
Possible pairs: \(\left(1,1),(1,2),(2,1),(2,2\right)\)
Products: \(1,2,2,4\)
Favourable cases: \(2\rightarrow 2\) cases out of 4
\(P=\frac{2}{4}=\frac{1}{2}\)

A 0.5

B 0.25

C 0.75

D 1

💡 Short Explanation

Use determinant product property. Count favourable determinant products. Probability equals favourable over total cases.

✅ Correct Answer Explanation

Given: \(∣AB∣=∣A∣∣B∣\) Possible determinant pairs are: \(\left(1,1),(1,2),(2,1),(2,2\right)\) Products become: \(1,2,2,4\) Two outcomes produce determinant 2. Total outcomes are four: \(P=\frac{2}{4}=\frac{1}{2}=0.5\) Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Counts only one favourable case instead of two.
Option C → Incorrectly assumes three favourable products equal 2.
Option D → Not all determinant products equal 2.

🎯 Strategy

Used: Substitution

Application:

List all determinant combinations and evaluate products systematically.

Final Logic:

Two favourable cases out of four give probability 0.5.

🧠 Memory Trick

"Product rule + counting cases."

📚 Class XII – Mathematics Part 1 – Chapter: Determinants – Topic: Product of Determinants

12 If \(AB=BA=I\), then \(B\) is:

A \(\operatorname{adj}\,A\)

B \(A^{2}\)

C \(A^{-1}\)

D \(I\)

💡 Short Explanation

Identity product defines inverse matrices. Left and right inverses coincide. \(B\) must equal inverse of \(A\).

✅ Correct Answer Explanation

By definition, if: \(AB=BA=I\) then \(B\) is the inverse of \(A\): \(AA^{-1}=A^{-1}A=I\) Thus \(B=A^{-1}\). Hence Option C is correct. Adjoint and identity matrices satisfy different properties.

❓ Why Other Options Are Incorrect

Option A → Adjoint alone does not necessarily produce identity.
Option B → Squaring a matrix is unrelated to inverse definition.
Option D → Identity matrix equals inverse only when \(A=I\).

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Use the standard matrix inverse definition directly.

Final Logic:

Identity multiplication characterizes inverses uniquely.

🧠 Memory Trick

"Product gives identity → inverse found."

📚 Class XII – Mathematics Part 1 – Chapter: Matrices – Topic: Inverse of a Matrix

13 Which are valid dimensional conditions?
I. \(A\) is \(n\times n\)
II. \(X\) is \(n\times 1\)
III. \(B\) is \(1\times n\)

A I only

B I and II

C II and III

D I, II, and III

💡 Short Explanation

Matrix equation is \(AX=B\). Dimensions must satisfy multiplication rules. \(B\) should be \(n\times 1\), not \(1\times n\).

✅ Correct Answer Explanation

For: \(AX=B\) if \(A\) is \(n\times n\) and \(X\) is \(n\times 1\), then product \(AX\) becomes \(n\times 1\). Therefore \(B\) must also be \(n\times 1\). Statement III is false because \(1\times n\) dimensions are incompatible. Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores the required column-vector dimension for \(X\).
Option C → Statement III is dimensionally incompatible.
Option D → \(B\) cannot be a row vector in the equation \(AX=B\).

🎯 Strategy

Used: Dimensional/Unit Analysis

Application:

Check matrix multiplication compatibility carefully.

Final Logic:

\(AX\) and \(B\) must have identical dimensions.

🧠 Memory Trick

"Matrix product dimensions must match."

📚 Class XII – Mathematics Part 1 – Chapter: Matrices – Topic: Matrix Equations

14 Correct logical steps:
1. Form \(AX=B\)
2. Premultiply: \(A^{-1}(AX)=A^{-1}B\)
3. Use associativity: \((A^{-1}A)X=IX\)
4. Conclude \(X=A^{-1}B\)

A 1, 2, 3, 4

B 4, 3, 2, 1

C 1, 4, 3, 2

D 1, 3, 4, 2

💡 Short Explanation

Start with matrix equation. Multiply by inverse matrix. Simplify using identity property.

✅ Correct Answer Explanation

To solve: \(AX=B\) premultiply both sides by \(A^{-1}\): \(A^{-1}(AX)=A^{-1}B\) Using associativity: \((A^{-1}A)X=IX\) Since: \(A^{-1}A=I\) we obtain: \(X=A^{-1}B\) Thus Option A gives the correct logical order.

❓ Why Other Options Are Incorrect

Option B → Reverses the derivation sequence completely.
Option C → Concludes solution before simplification steps are completed.
Option D → Associativity step must follow premultiplication, not precede it.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Follow the algebraic derivation process in exact sequence.

Final Logic:

Premultiplication precedes simplification and final conclusion.

🧠 Memory Trick

"Multiply inverse, simplify, conclude."

📚 Class XII – Mathematics Part 1 – Chapter: Matrices – Topic: Solving Matrix Equations

15 Assertion (A): A consistent system always has exactly one solution.
Reason (R): Infinite solutions indicate inconsistency.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Consistent systems may have infinite solutions. Infinite solutions still satisfy equations. Both statements are false.

✅ Correct Answer Explanation

A consistent system is one that has at least one solution. It may have either a unique solution or infinitely many solutions. Therefore Assertion is false. Infinite solutions still satisfy all equations simultaneously, so they indicate consistency, not inconsistency. Hence Reason is also false. Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Assertion is false because consistency does not imply uniqueness.
Option C → Infinite solutions do not imply inconsistency.
Option D → Reason is also false since infinite solutions remain consistent.

🎯 Strategy

Used: Extreme Word Filter

Application:

Check whether the word "always" creates an incorrect universal claim.

Final Logic:

Consistency includes both unique and infinite solutions.

🧠 Memory Trick

"Consistent means at least one solution."

📚 Class XII – Mathematics Part 1 – Chapter: Matrices – Topic: Consistent and Inconsistent Systems

16 Match the Following

List I	List II
1. \(∣A∣\neq 0\)	a. Inconsistent
2. \(∣A∣=0\) and \((adj⁡A)B\neq O\)	b. Consistent (infinite solutions)
3. \(∣A∣=0\) and \((adj⁡A)B=O\)	c. Unique solution

A 1–a, 2–b, 3–c

B 1–c, 2–a, 3–b

C 1–b, 2–a, 3–c

D 1–c, 2–b, 3–a

💡 Short Explanation

Nonzero determinant gives unique solution. Singular inconsistent systems fail compatibility. Zero determinant may still allow infinite solutions.

✅ Correct Answer Explanation

If: \(∣A∣\neq 0\) the system has a unique solution. If: \(∣A∣=0, (adj⁡A)B\neq O\) the system is inconsistent. If: \(∣A∣=0, (adj⁡A)B=O\) the system is consistent with infinitely many solutions. Hence matching becomes: 1–c, 2–a, 3–b.

❓ Why Other Options Are Incorrect

Option A → Unique solution and inconsistency conditions are interchanged incorrectly.
Option C → Nonzero determinant cannot produce infinitely many solutions.
Option D → Singular compatible systems are not inconsistent.

🎯 Strategy

Used: Option Grouping

Application:

Group determinant conditions into unique, inconsistent, and infinite-solution cases.

Final Logic:

Only nonzero determinant guarantees uniqueness.

🧠 Memory Trick

"Zero determinant needs compatibility check."

📚 Class XII – Mathematics Part 1 – Chapter: Matrices – Topic: System of Linear Equations

17 Three planes intersecting uniquely meet at:

A A line

B No point

C A single point

D Three lines

💡 Short Explanation

Unique intersection means exactly one common solution. Three planes correspond to three equations. One solution means one common point.

✅ Correct Answer Explanation

In three-dimensional geometry, each plane represents a linear equation. If three planes intersect uniquely, the corresponding system has exactly one solution. Geometrically, this means all three planes meet at one common point. Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → A line occurs when infinitely many common points exist.
Option B → No point indicates inconsistency among planes.
Option D → Three separate lines do not represent a unique common intersection.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Interpret algebraic uniqueness geometrically.

Final Logic:

Unique solution corresponds to one common intersection point.

🧠 Memory Trick

"One solution → one point."

📚 Class XII – Mathematics Part 1 – Chapter: Linear Equations – Topic: Geometric Interpretation

18 Identify incorrect statement when \(∣A∣=0\):

A \(A^{-1}\)does not exist

B System is always consistent

C \((adj⁡A)B\) determines consistency

D Unique solution is impossible

💡 Short Explanation

Singular systems may be consistent or inconsistent. Zero determinant prevents uniqueness. Compatibility condition determines consistency.

✅ Correct Answer Explanation

When: \(∣A∣=0\) the matrix is singular and inverse does not exist. A unique solution becomes impossible. However, the system may still be either consistent or inconsistent depending on: \((adj⁡A)B\) Therefore Option B is incorrect because consistency is not guaranteed.

❓ Why Other Options Are Incorrect

Option A → Singular matrices cannot possess inverses.
Option C → Compatibility conditions determine system behavior correctly.
Option D → Unique solutions require nonzero determinant.

🎯 Strategy

Used: Extreme Word Filter

Application:

Check the universal claim "always consistent" carefully.

Final Logic:

Singular systems may still be inconsistent.

🧠 Memory Trick

"Zero determinant ≠ always consistent."

📚 Class XII – Mathematics Part 1 – Chapter: Matrices – Topic: Singular Systems

19 Equations:
\(2x+3y=8,x-2y=-3\)
Coefficient matrix:
\(A=\left(\begin{pmatrix}2 & 3\\ 1 & -2\end{pmatrix}\right)∣A∣=2(-2)-3(1)=-4-3=-7\)

A 0

B 1

C -4

D -7

💡 Short Explanation

Use determinant formula \(ad-bc\). Multiply diagonal terms carefully. Subtract cross products.

✅ Correct Answer Explanation

For: \(A=\left(\begin{pmatrix}2 & 3\\ 1 & -2\end{pmatrix}\right)\) determinant: \(∣A∣=2(-2)-3(1)=-4-3=-7\) Hence Option D is correct. Other values result from incorrect arithmetic or omission of subtraction.

❓ Why Other Options Are Incorrect

Option A → Determinant does not vanish after correct calculation.
Option B → Incorrect simplification of determinant formula.
Option C → Omits subtraction of the second product term.

🎯 Strategy

Used: Substitution

Application:

Apply the 2×2 determinant formula directly.

Final Logic:

\(ad-bc=-7\).

🧠 Memory Trick

"Multiply diagonals, subtract cross."

📚 Class XII – Mathematics Part 1 – Chapter: Determinants – Topic: Determinant of Order 2

20 If \(AX=O\) and \(∣A∣\neq 0\), then:

A \(X=O\)(trivial solution)

B \(X\) undefined

C Infinite solutions

D \(X=A\)

💡 Short Explanation

Nonzero determinant implies invertibility. Premultiply by inverse matrix. Only zero vector solution remains.

✅ Correct Answer Explanation

Since: \(∣A∣\neq 0\) matrix \(A\) is invertible. From: \(AX=O\) premultiply by \(A^{-1}\): \(A^{-1}(AX)=A^{-1}OIX=OX=O\) Hence only the trivial solution exists. Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → \(X\) is fully determined after multiplying by inverse.
Option C → Infinite solutions occur only for singular matrices.
Option D → Solution vector cannot equal coefficient matrix.

🎯 Strategy

Used: Substitution

Application:

Use inverse multiplication to isolate the unknown vector.

Final Logic:

Invertible matrices allow only the trivial homogeneous solution.

🧠 Memory Trick

"Invertible homogeneous system → trivial solution."

📚 Class XII – Mathematics Part 1 – Chapter: Matrices – Topic: Homogeneous Systems

CUET UG Mathematics Booster Test 2 - Adjoint, Inverse Matrices and Applications of Determinants

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