CUET UG Applied Mathematics Booster Test 2 - Determinants and Their Properties -

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CUET UG Applied Mathematics Booster Test 2 - Determinants and Their Properties

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 2 - Determinants and Their Properties

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QUESTION 1 OF 20

If f(A) = k represents the determinant function mapping a matrix A to a number k, what strictly forms the domain (X) of this mapping function f?

Answer Review

1 If f(A) = k represents the determinant function mapping a matrix A to a number k, what strictly forms the domain (X) of this mapping function f?

A The set of all real numbers.

B The set of all rectangular matrices.

C The set of all square matrices.

D The set of non-singular matrices only.

💡 Short Explanation

�� Determinants are defined only for square matrices. �� Rectangular matrices do not possess determinants. �� Hence the domain consists of all square matrices.

✅ Correct Answer Explanation

The determinant is a scalar value associated only with square matrices. If a matrix is not square, the determinant cannot be evaluated. Option C is correct because the determinant function maps every square matrix to a scalar value. Option A is incorrect because real numbers are outputs (codomain), not inputs. Option B is incorrect because rectangular matrices do not have determinants. Option D is incorrect because even singular matrices have determinants; their determinant is simply zero.

❓ Why Other Options Are Incorrect

�� Option A → Real numbers represent determinant values, not the domain matrices.
�� Option B → Determinants are undefined for rectangular matrices.
�� Option D → Singular matrices are also square matrices and possess determinants equal to zero.

🎯 Strategy

Used: Elimination

Application: Remove options involving non-square matrices or outputs instead of inputs.

Final Logic: Since determinants exist only for square matrices, the domain must be all square matrices.

🧠 Memory Trick

"Determinant demands squares."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Definition of Determinant

2 Match the specific matrices to their corresponding determinant values based on the square matrix association framework.

List I	List II
1. Matrix Row 1: (4)	a. 5
2. Matrix Row 1: (2, 5), Row 2: (-1, 0)	b. 4
3. Matrix Row 1: (1, 0), Row 2: (0, 1)	c. 1
4. Matrix Row 1: (2, 3)	d. Determinant not defined

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

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

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

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

💡 Short Explanation

�� A 1×1 determinant equals its single element. �� A 2×2 determinant uses ad − bc. �� Non-square matrices do not have determinants.

✅ Correct Answer Explanation

(1): Determinant of [4] equals 4 → (b) (2): Determinant = (2×0) − (5×−1) = 5 → (a) (3): Identity matrix determinant = 1 → (c) (4): Matrix (2,3) is 1×2, not square → determinant undefined → (d) Thus, Option A is correct. Other options mismatch determinant calculations or matrix classifications.

❓ Why Other Options Are Incorrect

�� Option B → Incorrect determinant assignments for matrices A and B.
�� Option C → Incorrectly treats non-square matrix as determinant-defined.
�� Option D → Incorrectly matches determinant values and properties.

🎯 Strategy

Used: Substitution

Application: Compute each determinant directly and match systematically.

Final Logic: Correct determinant evaluations uniquely satisfy Option A.

🧠 Memory Trick

"1×1 → itself, 2×2 → ad − bc."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Evaluation of Determinants

3 If A is a 3 × 3 square matrix, which of the following logical statements hold true for computing its minors?
1. The minor of (a_{12}) involves evaluating a 2 × 2 determinant.
2. Minor (M_{31}) is computed by deleting the 3rd row and the 1st column.
3. The minor of an element is independent of the algebraic sign factor ((-1)^{i+j}).
4. A 2 × 3 matrix has perfectly defined minors for all elements.

A 1, 2, 3

B 2, 3, 4

C 1, 4

D 1, 2, 3, 4

💡 Short Explanation

�� A minor is obtained by deleting one row and one column. �� A cofactor includes the sign factor ((-1)^{i+j}). �� Minors are defined for square matrices in determinant theory.

✅ Correct Answer Explanation

For a 3 × 3 matrix, deleting one row and one column leaves a 2 × 2 determinant. Therefore, the minor of (a_{12}) is obtained by evaluating a 2 × 2 determinant. Hence, Statement 1 is correct. → The minor (M_{31}) is obtained by deleting the 3rd row and the 1st column and then evaluating the determinant of the remaining 2 × 2 matrix. Therefore, Statement 2 is correct. → The algebraic sign factor ((-1)^{i+j}) is used only while computing the cofactor: A₍ᵢⱼ₎ = ((-1)^{i+j}) M₍ᵢⱼ₎ Hence, Statement 3 is correct because the sign factor is not a part of the minor. → Statement 4 is incorrect because determinants are defined only for square matrices. Since a 2 × 3 matrix is not square, minors are not defined in the context of determinant theory. → Therefore, Statements 1, 2, and 3 are correct.

❓ Why Other Options Are Incorrect

�� Option B → Incorrect because it includes Statement 4, which is false.
�� Option C → Incorrect because it omits Statements 2 and 3, both of which are correct.
�� Option D → Incorrect because Statement 4 is false.

🎯 Strategy

Used

Elimination

Application:

�� Eliminate every option containing Statement 4 because determinant-based minors are defined only for square matrices.

Final Logic:

�� Minors are obtained by deleting a row and a column, while the sign factor belongs only to cofactors.

🧠 Memory Trick

"Minor Deletes; Cofactor Signs."

📚 A₍ᵢⱼ₎ = ((-1)^{i+j}) M₍ᵢⱼ₎

4 Identify the mathematically INCORRECT statement regarding minor evaluation:

A M_22 in a 3x3 matrix is found by expanding the remaining smaller 2x2 matrix.

B The minor M_ij is strictly a determinant itself.

C The minor of an element a_ij is always numerically and algebraically identical to the cofactor A_ij.

D Only elements within a square matrix have associated valid determinants/minors.

💡 Short Explanation

�� Cofactors include sign multipliers. �� Minors are determinants of reduced matrices. �� Minor and cofactor may differ in sign.

✅ Correct Answer Explanation

Option C is incorrect because a cofactor equals: A_(ij)=(-1)^(i+j)M_(ij) Thus minors and cofactors are not always identical. Option A is correct since deleting one row and column in a 3×3 matrix leaves a 2×2 determinant. Option B is correct because a minor itself is a determinant. Option D is correct because determinant concepts apply to square matrices.

❓ Why Other Options Are Incorrect

�� Option A → Correct definition of minor evaluation.
�� Option B → A minor is indeed a determinant of smaller order.
�� Option D → Determinants and minors are defined only for square matrices.

🎯 Strategy

Used: Odd One Out

Application: Identify the option contradicting the cofactor formula.

Final Logic: Only Option C ignores the sign factor in cofactors.

🧠 Memory Trick

"Cofactor = sign × minor."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Cofactors

5 In a mixture problem tracking determinant values, if the minor M_23 evaluates to 13 for a specific matrix element, what is the precisely defined cofactor A_23?

A 13

B -13

C 1/13

D -1/13

💡 Short Explanation

�� Cofactor formula uses sign multiplier. �� For position (2,3), i+j = 5. �� Odd sum gives negative sign.

✅ Correct Answer Explanation

The cofactor formula is: A_(ij)=(-1)^(i+j)M_(ij) For (2,3): 2+3=5 Since 5 is odd: (-1)^5=-1 Thus: A_(23)=-13 Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Ignores the negative sign due to odd index sum.
�� Option C → Cofactors are not reciprocals of minors.
�� Option D → Incorrect reciprocal form with wrong operation.

🎯 Strategy

Used: Substitution

Application: Apply cofactor sign formula directly.

Final Logic: Odd index sum gives negative cofactor.

🧠 Memory Trick

"Odd sum → negative cofactor."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Cofactor Formula

6 Under the strict constraint that an element's position yields an even sum of its row and column indices (i.e., i+j is an even integer), how does the cofactor A_ij compare algebraically to its minor M_ij?

A A_ij and M_ij possess opposite signs.

B A_ij is exactly algebraically equal to M_ij.

C A_ij is always strictly zero.

D M_ij becomes mathematically undefined.

💡 Short Explanation

�� Cofactor sign depends on i+j. �� Even sum gives positive sign. �� Hence cofactor equals minor.

✅ Correct Answer Explanation

The cofactor formula is: A_(ij)=(-1)^(i+j)M_(ij) If i+j is even: (-1)^(even)=+1 Therefore: A_(ij)=M_(ij) Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Opposite signs occur only for odd sums.
�� Option C → Cofactors are not necessarily zero.
�� Option D → Minor remains well-defined for square matrices.

🎯 Strategy

Used: Substitution

Application: Substitute even parity into the cofactor formula.

Final Logic: Even index sum produces positive sign.

🧠 Memory Trick

"Even sum → same sign."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Cofactors

7 Calculate the equated determinant integer (EMI) for the order 2 matrix modeled by Row 1: (-9, 2) and Row 2: (1, -7).
**Correct Answer: A) (-9 * -7) - (2 * 1) = 61**

A (-9 * -7) - (2 * 1) = 61

B (-9 * 1) - (-7 * 2) = 5

C (9 * 7) + 2 = 65

D (-9 * -7) + (2 * 1) = 65

💡 Short Explanation

�� Use determinant formula ad − bc. �� Multiply diagonal elements. �� Subtract cross product.

✅ Correct Answer Explanation

For matrix: ∣[-9, 2; 1, -7]∣ Determinant: (-9)(-7)-(2)(1)63-2=61 Hence Option A is correct.

❓ Why Other Options Are Incorrect

�� Option B → Uses incorrect determinant arrangement.
�� Option C → Uses wrong arithmetic operation.
�� Option D → Adds instead of subtracting cross product.

🎯 Strategy

Used: Substitution

Application: Directly apply ad − bc formula.

Final Logic: Determinant equals 61.

🧠 Memory Trick

"2×2 determinant = ad − bc."

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

8 In an order 3 matrix, if you calculate the moving average-like cross-sum: a11A31 + a12A32 + a13*A33, what is the resulting theoretical value?

A |A|

B -|A|

C 0

D 1

💡 Short Explanation

�� Expansion must use corresponding row cofactors. �� Different row-cofactor combinations give zero. �� This is a determinant property.

✅ Correct Answer Explanation

The expression: a_(11)A_(31)+a_(12)A_(32)+a_(13)A_(33) uses elements of the first row with cofactors of the third row. According to determinant properties, when elements of one row are multiplied with cofactors of another distinct row, the sum equals zero. Thus, Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Determinant arises only with matching cofactors from the same row/column.
�� Option B → No negative determinant property applies here.
�� Option D → Such expressions do not simplify to 1 generally.

🎯 Strategy

Used: Conceptual/Tonal Matching

Application: Match the expression with determinant expansion identities.

Final Logic: Different-row cofactor expansion gives zero.

🧠 Memory Trick

"Different row + cofactors = zero."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Properties of Cofactors

9 Given a square matrix A with a determinant |A| = 42, what is the probability-based certainty of the value of |A'| (the transpose of matrix A)?

A -42

B 1/42

C 42

D 0

💡 Short Explanation

�� Determinant remains unchanged under transpose. �� |A| = |A′|. �� Hence transpose preserves value.

✅ Correct Answer Explanation

A determinant has the property: ∣A^'∣=∣A∣ Since: ∣A∣=42 therefore: ∣A^'∣=42 Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Transpose does not change sign automatically.
�� Option B → Determinant is not inverted under transpose.
�� Option D → Determinant remains unchanged, not zero.

🎯 Strategy

Used: Direct Formula Application

Application: Apply transpose property directly.

Final Logic: Determinant remains invariant under transpose.

🧠 Memory Trick

"Transpose preserves determinant."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Properties of Determinants

10 P10 VECTORS If vector row R1 is interchanged directly with vector row R3 in a matrix A (where initially |A| = 15), what is the newly formed determinant value?

A 15

B -15

C 0

D 30

💡 Short Explanation

�� Interchanging two rows changes sign. �� Magnitude remains same. �� Determinant becomes negative.

✅ Correct Answer Explanation

One fundamental determinant property states: If two rows (or columns) are interchanged, the determinant changes sign. Initially: ∣A∣=15 After interchange: ∣A∣=-15 Thus Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Ignores sign change property.
�� Option C → Determinant becomes zero only for identical/proportional rows.
�� Option D → Row interchange does not double determinant.

🎯 Strategy

Used: Conceptual/Tonal Matching

Application: Recall determinant sign-change property under row interchange.

Final Logic: Row swap changes only the sign of determinant.

🧠 Memory Trick

"Swap rows → flip sign."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Properties of Determinants

11 If a coordinate matrix yields a determinant area calculation of exactly 0 (indicating that the 3 points are structurally collinear), the matrix is mathematically defined as:

A Skew-symmetric

B Singular

C Non-singular

D Identity

💡 Short Explanation

A matrix with determinant 0 is called singular. Collinear points give zero area. Zero determinant implies non-invertibility.

✅ Correct Answer Explanation

A matrix is called singular when its determinant equals zero. Here, the determinant used in the area calculation becomes 0 because the points are collinear. In determinant theory, if ∣A∣=0, then the matrix does not possess an inverse and is classified as singular. Option B is correct because singular matrices always have determinant zero. Option A is incorrect because skew-symmetric matrices satisfy A^'=-A. Option C is incorrect because non-singular matrices require ∣A∣≠0. Option D is incorrect because identity matrices have determinant 1.

❓ Why Other Options Are Incorrect

Option A → Skew-symmetric matrices are defined using transpose properties, not determinant zero.
Option C → Non-singular matrices must have a non-zero determinant.
Option D → Identity matrices always have determinant equal to 1.

🎯 Strategy

Used: Elimination

Application: The determinant being exactly zero directly eliminates all options except the singular matrix condition.

Final Logic: ∣A∣=0⇒matrix is singular.

🧠 Memory Trick

"Zero determinant = Singular certainty."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Singular and Non-Singular Matrices

12 For a matrix P to possess a valid inverse (P^-1), which integral condition must its determinant |P| firmly satisfy to be deemed non-singular?

A |P| = 1 strictly

B |P| = 0

C |P| != 0

D |P| must be a positive integer

💡 Short Explanation

Inverse exists only for non-singular matrices. Non-singular means determinant is non-zero. Determinant cannot be zero.

✅ Correct Answer Explanation

A square matrix possesses an inverse only if its determinant is non-zero. Mathematically: A^(-1) exists ⟺ ∣A∣≠0 Option C is correct because a non-zero determinant guarantees invertibility. Option A is incorrect because determinant need not be exactly 1. Option B is incorrect because determinant zero makes the matrix singular. Option D is incorrect because the determinant can be negative or fractional.

❓ Why Other Options Are Incorrect

Option A → Determinant 1 is sufficient but not necessary.
Option B → Zero determinant implies no inverse exists.
Option D → Determinants are not restricted to positive integers.

🎯 Strategy

Used: Elimination

Application: Use the inverse existence condition from determinant theory to remove incorrect constraints.

Final Logic: Inverse exists only when determinant is non-zero.

🧠 Memory Trick

"Non-zero means inverse flows."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Inverse of a Matrix Using Determinants

13 If the first and third columns of a 3x3 determinant are completely identical in every element, the final evaluated value of the determinant is:

A Doubled

B Undefined

C Zero

D One

💡 Short Explanation

Two identical rows/columns make determinant zero. Determinants depend on linear independence. Identical columns imply dependence.

✅ Correct Answer Explanation

One of the fundamental properties of determinants states: If any two rows or columns are identical, the determinant equals zero. Since the first and third columns are identical here, the determinant must vanish. Option C is correct. Option A is incorrect because determinants do not double due to identical columns. Option B is incorrect because the determinant is well-defined. Option D is incorrect because the value is not necessarily 1.

❓ Why Other Options Are Incorrect

Option A → Identical columns cause determinant collapse, not doubling.
Option B → Determinant exists for square matrices.
Option D → No determinant property forces value 1 here.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: The phrase "completely identical" directly signals the determinant-zero property.

Final Logic: Identical columns ⇒ determinant equals 0.

🧠 Memory Trick

"Same rows, determinant goes."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Properties of Determinants

14 If a matrix represents a system where an entire row is composed exclusively of zeros, evaluating its determinant strictly categorizes the matrix as:

A A singular matrix (|A| = 0)

B A non-singular matrix (|A| != 0)

C An identity matrix (|A| = 1)

D An inverse matrix

💡 Short Explanation

Zero row gives determinant zero. Zero determinant implies singularity. Such matrices are non-invertible.

✅ Correct Answer Explanation

A determinant becomes zero if any row or column contains all zeros. Thus: ∣A∣=0 Hence, the matrix is singular. Option A is correct. Option B is incorrect because non-singular matrices require non-zero determinant. Option C is incorrect because identity matrices have ones on the diagonal. Option D is incorrect because inverse existence fails.

❓ Why Other Options Are Incorrect

Option B → Non-singular matrices cannot have zero determinant.
Option C → Identity matrices never contain a complete zero row.
Option D → Singular matrices do not possess inverses.

🎯 Strategy

Used: Elimination

Application: A full zero row immediately implies determinant zero.

Final Logic: Zero row ⇒ determinant 0 ⇒ singular matrix.

🧠 Memory Trick

"Zero row, inverse no."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Properties of Determinants

15 If Row 1 elements are exactly proportional to Row 2 elements by a scalar factor k (e.g., R1 = k * R2), the determinant will evaluate to:

A k

B 1/k

C 0

D k^2

💡 Short Explanation

Proportional rows are linearly dependent. Linear dependence makes determinant zero. Determinant measures independence.

✅ Correct Answer Explanation

If one row is a scalar multiple of another row, the rows are linearly dependent. A determinant with proportional rows or columns always evaluates to zero. Option C is correct. Option A and Option D incorrectly treat determinant as directly scaling. Option B has no determinant basis.

❓ Why Other Options Are Incorrect

Option A → Determinant does not equal proportionality constant.
Option B → Reciprocal relation is unrelated here.
Option D → No squaring rule exists for proportional rows.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: The keyword "proportional" directly connects to determinant-zero property.

Final Logic: Proportional rows ⇒ determinant vanishes.

🧠 Memory Trick

"Proportional rows kill determinant."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Properties of Determinants

16 If the linear operation C1 -> C1 + 5C2 is applied to a matrix bearing an initial determinant D, the new determinant will mathematically evaluate to:

A 5D

B D + 5

C D

D -D

💡 Short Explanation

Adding a multiple of one column to another does not change determinant. Determinant remains invariant. Elementary operation preserves value.

✅ Correct Answer Explanation

One important determinant property states: Adding a multiple of one row/column to another row/column leaves the determinant unchanged. Thus: C_1→C_1+5C_2 does not affect determinant value. Option C is correct. Option A incorrectly assumes multiplication scaling. Option B is not a determinant property. Option D applies only to row interchange.

❓ Why Other Options Are Incorrect

Option A → Determinant changes only when an entire row/column is multiplied.
Option B → Determinants are not altered additively this way.
Option D → Sign change occurs during interchange, not addition operation.

🎯 Strategy

Used: Substitution

Application: Apply the standard determinant operation rule directly.

Final Logic: Row/column replacement by addition leaves determinant unchanged.

🧠 Memory Trick

"Add rows freely, determinant stays."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Elementary Operations on Determinants

17 If a 3x3 matrix A has a determinant of 10, and a scalar multiple k=3 is multiplied into every single element of matrix A, what is the determinant of the newly formed matrix?

A 30

B 90

C 270

D 10

💡 Short Explanation

For an n×n matrix, determinant scales by k^n. Here n=3. So determinant becomes 3^3×10.

✅ Correct Answer Explanation

If every element of an n×n matrix is multiplied by scalar k, then: ∣kA∣=k^n∣A∣ For a 3×3 matrix: ∣3A∣=3^3×10=27×10=270 Option C is correct. Option A incorrectly multiplies only once. Option B incorrectly uses 3^2. Option D ignores scaling.

❓ Why Other Options Are Incorrect

Option A → Determinant scaling is not linear for 3×3 matrices.
Option B → Uses square scaling instead of cube scaling.
Option D → Determinant definitely changes after scalar multiplication.

🎯 Strategy

Used: Substitution

Application: Substitute into the determinant scaling formula k^n∣A∣.

Final Logic: 3^3×10=270.

🧠 Memory Trick

"n×n matrix ⇒ kⁿ scaling."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Multiplication Property of Determinants

18 Given two 2x2 square matrices A and B, if |A| = -2 and |B| = 6, what is the numerical value of the combined product determinant |AB|?

A 4

B -12

C -3

D 12

💡 Short Explanation

Determinant of product equals product of determinants. Multiply determinant values directly. (-2)(6)=-12.

✅ Correct Answer Explanation

For square matrices: ∣AB∣=∣A∣∣B∣ Given: ∣A∣=-2,∣B∣=6 Therefore: ∣AB∣=(-2)(6)=-12 Option B is correct. Option A, C, and D do not satisfy the determinant product property.

❓ Why Other Options Are Incorrect

Option A → Incorrect multiplication result.
Option C → Does not follow determinant product rule.
Option D → Ignores the negative sign of ∣A∣.

🎯 Strategy

Used: Substitution

Application: Directly apply the determinant multiplication property.

Final Logic: ∣AB∣=∣A∣∣B∣=-12.

🧠 Memory Trick

"Product matrices → product determinants."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Determinant of Product of Matrices

📄 Passage

According to advanced determinant properties, if some or all elements of a specific row or column of a determinant are expressed as a mathematical sum of two or more terms, then the determinant can be split and explicitly expressed as a sum of two or more separate determinants. This split strictly applies only to the row or column containing the sum, while all remaining identical rows or columns are heavily preserved across all the newly formed determinants to maintain equality.

Question:
If a single column in a 3x3 matrix is expressed as the sum of three algebraic terms, how many distinct determinants can the original determinant be split into?

A 1

B 2

C 3

D 9

💡 Short Explanation

Determinants are linear in rows/columns. Sum of three terms gives three determinants. Splitting preserves equality.

✅ Correct Answer Explanation

The determinant summation property states: If elements of one row/column are expressed as sums, the determinant splits into separate determinants corresponding to each term. Since the column contains three summed terms, the determinant splits into three determinants. Option C is correct. Option A ignores splitting. Option B applies only to two-term sums. Option D incorrectly multiplies possibilities.

❓ Why Other Options Are Incorrect

Option A → Summation property explicitly creates multiple determinants.
Option B → Applicable only for binomial splitting.
Option D → No quadratic expansion occurs.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: The phrase "sum of three terms" directly indicates three determinant components.

Final Logic: Three summed terms ⇒ three split determinants.

🧠 Memory Trick

"Three terms → three determinants."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Linearity Property of Determinants

📄 Passage

Question:
During matrix splitting via the summation property, what must logically happen to the rows or columns that do NOT contain the summed terms?

A They are divided evenly among the new determinants.

B They are duplicated exactly in all the newly formed determinants.

C They are systematically replaced by zeros.

D They change their algebraic signs.

💡 Short Explanation

Only the summed row/column changes. Remaining rows/columns stay identical. This preserves determinant equality.

✅ Correct Answer Explanation

Under the determinant summation property, splitting occurs only in the row or column containing the sums. All remaining rows and columns remain unchanged and are duplicated exactly across all newly formed determinants. Option B is correct. Option A is incorrect because rows are not divided. Option C is incorrect because no zero replacement occurs. Option D is incorrect because signs remain unchanged.

❓ Why Other Options Are Incorrect

Option A → Determinant splitting preserves full structure.
Option C → Zero substitution is not part of the property.
Option D → Algebraic signs do not automatically change.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Focus on the phrase "do NOT contain the summed terms."

Final Logic: Only summed row/column changes; others remain identical.

🧠 Memory Trick

"Only split row changes."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Sum Property of Determinants

CUET UG Applied Mathematics Booster Test 2 - Determinants and Their Properties

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