CUET UG Applied Mathematics Booster Test 2 - Matrix Algebra and Transformations -

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CUET UG Applied Mathematics Booster Test 2 - Matrix Algebra and Transformations

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 2 - Matrix Algebra and Transformations

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QUESTION 1 OF 20

If the integral values of a matrix A are scaled by a fractional scalar k = 1/2, what happens to the order of the newly formed matrix 1/2 A compared to the original matrix A?

Answer Review

1 If the integral values of a matrix A are scaled by a fractional scalar k = 1/2, what happens to the order of the newly formed matrix 1/2 A compared to the original matrix A?

A The order is halved.

B The order is squared.

C The order remains unchanged.

D The rows are halved, but columns remain the same.

💡 Short Explanation

�� Scalar multiplication changes only matrix entries. �� Number of rows and columns stays fixed. �� Matrix order is unaffected by scalar values.

✅ Correct Answer Explanation

In scalar multiplication, every element of matrix A=[a_(ij)]is multiplied by the scalar k. → The structure of the matrix does not change during this process. Hence, if A is of order m×n, then 1/2A is also of order m×n. → Option C is correct because only element values change, not dimensions. → Option A is incorrect because the order cannot be numerically halved. → Option B is incorrect because scalar multiplication does not square dimensions. → Option D is incorrect because neither rows nor columns are altered individually.

❓ Why Other Options Are Incorrect

�� Option A → Matrix dimensions are independent of scalar size.
�� Option B → Squaring applies to numbers, not matrix order in scalar multiplication.
�� Option D → Scalar multiplication affects entries only, not row-column count.

🎯 Strategy

Used: Elimination

Application: Eliminate options suggesting dimensional change because scalar multiplication only scales entries.

Final Logic: Since rows and columns remain identical, the order remains unchanged.

🧠 Memory Trick

"Scalar scales values, not size."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Scalar Multiplication of Matrices

2 In modifying an area coordinate matrix A using scalars p and q, the distributive relation (p + q)A is expanded. If p = 2, q = -2, what is the result of (p + q)A?

A 4A

B A

C O (Zero matrix)

D -4A

💡 Short Explanation

�� p+q=2+(-2)=0 �� 0A=O �� Zero scalar gives zero matrix.

✅ Correct Answer Explanation

Using distributive property: (p+q)A=(2+(-2))A=0A → Multiplying any matrix by scalar 0 gives the zero matrix. → Hence the result is the zero matrix O. → Option C is correct. → Option A is incorrect because 2+(-2)≠4. → Option B is incorrect because multiplication by 0 cannot return the original matrix. → Option D is incorrect because no negative scalar sum of -4 appears.

❓ Why Other Options Are Incorrect

�� Option A → Incorrect scalar addition.
�� Option B → Zero multiplication cannot preserve the matrix.
�� Option D → Scalar value is not -4.

🎯 Strategy

Used: Substitution

Application: Directly substitute p=2 and q=-2.

Final Logic: Since p+q=0, the result is the zero matrix.

🧠 Memory Trick

"Zero scalar → Zero matrix."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Scalar Multiplication Properties =

3 Two row matrices
V₁ = [x 4]
and
V₂ = [2 y]
are added to obtain the matrix
S = [5 9].
What are the values of x and y?

A x = 2, y = 4

B x = 3, y = 5

C x = 7, y = 13

D x = 5, y = 9

💡 Short Explanation

�� Matrix addition is performed element-wise. �� Corresponding elements of equal matrices are compared. �� Solve the resulting linear equations.

✅ Correct Answer Explanation

Matrix addition is carried out by adding the corresponding elements of the matrices. Given, V₁ = [x 4] V₂ = [2 y] S = [5 9] Using matrix addition, V₁ + V₂ = S ⇒ [x + 2 4 + y] = [5 9] Equating the corresponding elements, x + 2 = 5 ⇒ x = 3 4 + y = 9 ⇒ y = 5 Therefore, x = 3 and y = 5. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → For x = 2 and y = 4, the sum becomes [4 8], not [5 9].
�� Option C → For x = 7 and y = 13, the sum becomes [9 17], which is incorrect.
�� Option D → For x = 5 and y = 9, the sum becomes [7 13], which does not match the given matrix.

🎯 Strategy

Used

Substitution

Application:

�� Add the corresponding elements and compare them with the resultant matrix to form equations.

Final Logic:

�� Matrix equality requires corresponding elements to be equal; solving these equations gives x = 3 and y = 5.

🧠 Memory Trick

"Equal Matrices → Equal Positions."

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Equality and Addition of Matrices

4 If you randomly select a matrix to act as the additive inverse of matrix A = [2 -1 ; 3 0], what is the absolute probability that it strictly equals [-2 1 ; -3 0] to yield a zero matrix sum?

A 0.5

B 1

C 0

D 0.25

💡 Short Explanation

�� Additive inverse is unique. �� A+(-A)=O �� Only one exact inverse exists.

✅ Correct Answer Explanation

The additive inverse of matrix A is obtained by changing signs of all elements. -A=[[-2, 1; -3, 0]] → Adding A and -A gives the zero matrix. → Since additive inverse is unique, the probability is certain. → Hence Option B is correct. → Options A, C, and D incorrectly treat the inverse as uncertain or partial.

❓ Why Other Options Are Incorrect

�� Option A → Additive inverse is not a probabilistic half-case.
�� Option C → The given matrix is exactly the inverse.
�� Option D → There is no one-fourth uncertainty involved.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: "Absolute probability" and exact inverse imply certainty.

Final Logic: Exact additive inverse always gives the zero matrix.

🧠 Memory Trick

"Inverse cancels perfectly."

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

5 A 5-day moving average dataset tracks prices P1 and P2 in matrices. To analytically find the difference matrix P1 - P2, what must mathematically hold true about their structures?

A They must both be column matrices.

B They must strictly share the exact same order m × n.

C One must be a transpose of the other.

D P1 must have more rows than P2.

💡 Short Explanation

�� Matrix subtraction is element-wise. �� Corresponding positions must exist. �� Same order is mandatory.

✅ Correct Answer Explanation

Matrix subtraction is defined only when both matrices have identical dimensions. → Corresponding entries are subtracted position-wise. → Therefore P_1 and P_2 must both be of order m×n. → Option B is correct. → Option A is unnecessary because subtraction works for any same-order matrices. → Option C is incorrect because transpose changes dimensions. → Option D is unrelated to subtraction rules.

❓ Why Other Options Are Incorrect

�� Option A → Column form is not compulsory.
�� Option C → Transpose relation is not required.
�� Option D → Different row counts make subtraction undefined.

🎯 Strategy

Used: Elimination

Application: Remove options imposing unnecessary structural restrictions.

Final Logic: Same order alone guarantees valid subtraction.

🧠 Memory Trick

"Subtract only same-sized matrices."

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

6 An EMI deficit matrix is modeled as the negative of matrix A, denoted as -A. If A = [-4 6], what is the exact element representation of -A?

A [4 -6]

B [-4 -6]

C [6 -4]

D [1/4 -1/6]

💡 Short Explanation

�� Negative matrix changes signs. �� Multiply each entry by -1. �� Order remains same.

✅ Correct Answer Explanation

The negative of a matrix is obtained by multiplying every element by -1. A=[-4 6]-A=[4 -6] → Therefore Option A is correct. → Option B changes only one sign incorrectly. → Option C rearranges entries. → Option D takes reciprocals instead of negatives.

❓ Why Other Options Are Incorrect

�� Option B → Sign of -4 not changed correctly.
�� Option C → Matrix elements are reordered incorrectly.
�� Option D → Negatives are not reciprocals.

🎯 Strategy

Used: Substitution

Application: Multiply each element directly by -1.

Final Logic: Sign reversal gives [4 , 6].

🧠 Memory Trick

"Minus matrix = flip all signs."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Negative of a Matrix

7 For a constraint region tracking product demand, matrix C of order 3 × 2 is multiplied by matrix D. What must be the exact number of rows in D for the product CD to be mathematically defined?

A 3

B 2

C 1

D 6

💡 Short Explanation

�� Matrix multiplication requires inner dimensions equal. �� Columns of first = rows of second. �� 3×2 needs second matrix rows = 2.

✅ Correct Answer Explanation

For multiplication CD to exist: (columns of C)=(rows of D) → Since matrix C is of order 3×2, it has 2 columns. → Therefore matrix D must have exactly 2 rows. → Hence Option B is correct. → Other options violate matrix multiplication compatibility conditions.

❓ Why Other Options Are Incorrect

�� Option A → 3 rows are not required.
�� Option C → One row makes multiplication undefined.
�� Option D → Number 6 has no compatibility relevance.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Match inner dimensions for valid multiplication.

Final Logic: Second matrix rows must equal first matrix columns.

🧠 Memory Trick

"Inside numbers must match."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Matrix Multiplication Compatibility

8 A chemical mixture equation requires the matrix product AB. If A is of order 2 × 3 and B is of order 3 × 4, what is the resulting order of the mixture product matrix AB?

A 3 × 3

B 4 × 2

C 2 × 4

D 2 × 3

💡 Short Explanation

�� Product order uses outer dimensions. �� Inner dimensions must match. �� 2×3(3×4) gives 2×4.

✅ Correct Answer Explanation

Matrix multiplication rule: (m×n)(n×p)=(m×p) → Here: (2×3)(3×4)=(2×4) → Therefore the resulting matrix order is 2×4. → Option C is correct. → Other options incorrectly use inner dimensions or reverse dimensions.

❓ Why Other Options Are Incorrect

�� Option A → Uses inner dimensions incorrectly.
�� Option B → Reverses the order improperly.
�� Option D → Retains first matrix order incorrectly.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Use matrix multiplication dimension rule directly.

Final Logic: Outer dimensions determine the product order.

🧠 Memory Trick

"Outer survives, inner matches."

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

9 Identify the INCORRECT analytical statement concerning distributive laws in matrix algebra:

A A(B +

B C = AC + BC is valid if orders are compatible.

C = AB - AC is fully supported by distributive principles.

D A(B -

💡 Short Explanation

�� Matrix multiplication is generally non-commutative. �� Correct distributive laws preserve order. �� Reversing multiplication order changes results.

✅ Correct Answer Explanation

Matrix multiplication generally does NOT satisfy commutative property: AB≠BA → Therefore: A(B+C)=AB+AC not BA+CA. → Option C is incorrect because it wrongly assumes universal commutativity. → Options A, B, and D are valid distributive laws when orders are compatible.

❓ Why Other Options Are Incorrect

�� Option A → Correct left distributive property.
�� Option B → Correct right distributive property.
�� Option D → Valid distributive law for subtraction.

🎯 Strategy

Used: Extreme Word Filter

Application: The phrase "always true" signals danger because matrix multiplication is not universally commutative.

Final Logic: Reversing multiplication order invalidates the distributive relation.

🧠 Memory Trick

"Matrices distribute, not commute."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Properties of Matrix Multiplication

10 Which of the following analytical scenarios accurately depict the non-commutative nature of general matrix multiplication (AB ≠ BA)?
1. AB is defined, but BA is not structurally defined.
2. AB and BA are both defined, but they form matrices of different orders.
3. AB and BA are both defined and are of the same order, but their corresponding elements are unequal.
4. Matrix multiplication is always commutative.

A 1, 2, 4

B 1, 2

C 1, 2, 3

D 1, 2, 3, 4

💡 Short Explanation

�� Matrix multiplication is generally non-commutative. �� The order of multiplication affects the result. �� Products may differ in existence, order, or entries.

✅ Correct Answer Explanation

Matrix multiplication is not commutative in general. That is, AB ≠ BA → Statement 1 is correct because there are cases where the product AB is defined, but BA is not defined due to incompatible matrix orders. Example: If A is of order 2 × 3 and B is of order 3 × 4, then AB is of order 2 × 4, but BA is not defined. → Statement 2 is also correct because both AB and BA may be defined but have different orders. Example: If A is of order 2 × 3 and B is of order 3 × 2, then AB is of order 2 × 2, while BA is of order 3 × 3. Thus, both products exist but have different orders. → Statement 3 is correct because even when both AB and BA are defined and have the same order (for example, when A and B are square matrices of the same order), their corresponding elements are generally different. Example: AB ≠ BA even though both are 2 × 2 matrices. → Statement 4 is incorrect because matrix multiplication is not always commutative. → Therefore, Statements 1, 2, and 3 are correct.

❓ Why Other Options Are Incorrect

�� Option A → Incorrect because it includes Statement 4, which is false.
�� Option B → Incorrect because Statement 3 is also a valid property of non-commutativity.
�� Option D → Incorrect because Statement 4 is false.

🎯 Strategy

Used

Option Grouping

Application:

�� Evaluate each statement independently and eliminate every option containing the false statement that matrix multiplication is always commutative.

Final Logic:

�� Matrix multiplication may differ in definition, order, or numerical values depending on the order of multiplication.

🧠 Memory Trick

"Order Matters: AB ≠ BA"

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Multiplication of Matrices and Non-Commutativity

11 Match the algebraic matrix equations to their simplified forms, given I is the identity matrix and A is a square matrix:

List I	List II
1. A × I	a. O (Zero matrix)
2. I × A	b. I (Identity matrix)
3. I × I	c. IA = A
4. A × O	d. AI = A

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

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

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

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

💡 Short Explanation

�� Identity matrix leaves a matrix unchanged. �� Zero matrix annihilates multiplication. �� Identity multiplied by itself remains identity.

✅ Correct Answer Explanation

Matrix multiplication properties give: AI=A and IA=A because identity matrix acts like multiplicative identity. → Also: I×I=I and A×O=O where O is the zero matrix. → Therefore: (1) A × I → A → (d) (2) I × A → A → (c) (3) I × I → I → (b) (4) A × O → O → (a) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Gives wrong result for I×I.
�� Option B → Incorrectly matches identity and zero properties.
�� Option D → Incorrectly treats identity multiplication as zero multiplication.

🎯 Strategy

Used: Option Grouping

Application: Apply standard identity and zero matrix multiplication rules systematically.

Final Logic: Identity preserves matrices, while zero matrix produces zero product.

🧠 Memory Trick

"I keeps same, O makes zero."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Identity and Zero Matrix Properties

12 If A = [0 -9 ; 0 4] and B = [1 -5 ; 0 0], evaluate the product AB. What type of matrix is the result?

A Identity Matrix

B Zero Matrix (O)

C Symmetric Matrix

D Diagonal Matrix

💡 Short Explanation

�� Multiply corresponding rows and columns. �� Every entry evaluates to zero. �� Resulting matrix becomes zero matrix.

✅ Correct Answer Explanation

Compute: AB=[[0, -9; 0, 4]][[1, -5; 0, 0]] First row: 0(1)+(-9)(0)=0 0(-5)+(-9)(0)=0 Second row: 0(1)+4(0)=0 0(-5)+4(0)=0 Thus, AB=[[0, 0; 0, 0]] which is the zero matrix.

❓ Why Other Options Are Incorrect

�� Option A → Identity matrix requires 1s on diagonal.
�� Option C → Although zero matrix is symmetric, the most precise classification here is zero matrix.
�� Option D → Diagonal matrix may have non-zero diagonal entries; here all entries are zero.

🎯 Strategy

Used: Substitution

Application: Directly multiply rows and columns to determine matrix type.

Final Logic: All product entries become zero, giving a zero matrix.

🧠 Memory Trick

"Zero everywhere ⇒ Zero matrix."

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

13 If A is a rectangular matrix of order 3 × 2, and A' is its transpose, what mathematical transformation happens to the general element a_ij of A?

A It becomes a_ij in A'.

B It becomes -a_ij in A'.

C It becomes a_ji in A'.

D It is squared.

💡 Short Explanation

�� Transpose interchanges rows and columns. �� Element positions swap indices. �� a_(ij)becomes a_(ji).

✅ Correct Answer Explanation

By definition of transpose: (A^')_(ij), a_(ji) → Thus every element changes position by interchanging row and column indices. → Example: If a_(12)is in first row and second column of A, it becomes a_(21)in transpose. → Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Elements do not remain at same coordinates.
�� Option B → Transpose does not change signs.
�� Option D → Transpose does not square elements.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Analyze positional transformation of indices during transpose.

Final Logic: Transpose swaps row index and column index.

🧠 Memory Trick

"Transpose = Index Swap."

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

14 Analytically applying transpose properties, what does the expression ((3A) + B)' simplify to?

A 3A' + B'

B A' + 3B'

C 3A + B

D 1/3 A' + B'

💡 Short Explanation

�� Transpose distributes over addition. �� Scalar remains unchanged during transpose. �� Apply transpose separately to each matrix.

✅ Correct Answer Explanation

Using transpose properties: (X+Y)^'=X^'+Y^' and (kA)^', kA^' Therefore: ((3A)+B)^'=(3A)^'+B^'=3A^'+B^' Hence Option A is correct.

❓ Why Other Options Are Incorrect

�� Option B → Scalar applies to A, not B.
�� Option C → Transpose operation ignored entirely.
�� Option D → Scalar 3 does not become 1/3.

🎯 Strategy

Used: Substitution

Application: Apply transpose rules step-by-step.

Final Logic: Transpose distributes and scalar remains unchanged.

🧠 Memory Trick

"Transpose spreads across plus."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Properties of Transpose

15 If matrix A = [1 x ; 4 5] is definitively a symmetric matrix, what must be the strict value of x?

A 1

B 4

C 5

D -4

💡 Short Explanation

�� Symmetric matrices satisfy A^'=A. �� Corresponding off-diagonal entries are equal. �� Thus x=4.

✅ Correct Answer Explanation

For a symmetric matrix: a_(ij)=a_(ji) Given: A=[[1, x; 4, 5]] Off-diagonal elements must match: x=4 Thus Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Does not satisfy symmetry condition.
�� Option C → Off-diagonal elements become unequal.
�� Option D → Negative value violates equality condition.

🎯 Strategy

Used: Substitution

Application: Compare corresponding off-diagonal entries directly.

Final Logic: Symmetric matrices require equal mirrored elements.

🧠 Memory Trick

"Symmetric means mirror-equal."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Symmetric Matrices

16 To computationally verify that matrix M is symmetric, which structural equivalence is systematically checked across all indices?

A m_ij = m_ij

B m_ij = -m_ji

C m_ij = m_ji for all i and j

D m_ii = 0

💡 Short Explanation

�� Symmetry compares mirrored entries. �� Row-column interchange should not change values. �� Condition holds for all indices.

✅ Correct Answer Explanation

A matrix is symmetric if: M^'=M which means: m_(ij)=m_(ji) for every valid pair of indices. → Option B defines skew-symmetric matrices. → Option D is only a skew-symmetric diagonal property.

❓ Why Other Options Are Incorrect

�� Option A → Trivial equality provides no symmetry condition.
�� Option B → Represents skew-symmetry.
�� Option D → Symmetric matrices need not have zero diagonals.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match formal definition of symmetric matrices.

Final Logic: Symmetry requires equality across transpose positions.

🧠 Memory Trick

"Symmetric = Same across diagonal."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Symmetric Matrix Definition

17 If matrix B is skew-symmetric, and element b_12 = 7, what is the exact numerical value of the corresponding transposed element b_21?

A 7

B 1/7

C -7

D 0

💡 Short Explanation

�� Skew-symmetric matrices satisfy B^'=-B. �� Corresponding entries are negatives. �� Thus b_(21)=-b_(12).

✅ Correct Answer Explanation

For skew-symmetric matrices: b_(ij)=-b_(ji) Given: b_(12)=7 Therefore: b_(21)=-7 Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Same sign corresponds to symmetric matrices.
�� Option B → Reciprocal relationship is unrelated.
�� Option D → Only diagonal entries are necessarily zero.

🎯 Strategy

Used: Substitution

Application: Substitute given value into skew-symmetric relation.

Final Logic: Opposite sign appears across transpose positions.

🧠 Memory Trick

"Skew flips the sign."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Skew-Symmetric Matrices

18 Why must the diagonal elements of a skew-symmetric matrix A = [a_ij] be explicitly zero?

A Because a_ij = a_ji, implying a_ii = 1.

B Because a_ij = -a_ji requires a_ii = -a_ii, which is only possible if a_ii = 0.

C Because scalar matrices require zeros.

D Because identity matrices dictate diagonal zeros.

💡 Short Explanation

�� Skew-symmetry gives a_(ii)=-a_(ii). �� Only zero satisfies this equation. �� Hence all diagonal entries are zero.

✅ Correct Answer Explanation

In a skew-symmetric matrix: a_(ij)=-a_(ji) For diagonal elements, put i=j: a_(ii)=-a_(ii) Adding both sides: 2a_(ii)=0 Thus: a_(ii)=0 Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → This is symmetric matrix condition, not skew-symmetric.
�� Option C → Scalar matrices need not have zero diagonals.
�� Option D → Identity matrices have diagonal 1s, not 0s.

🎯 Strategy

Used: Substitution

Application: Substitute i=j into skew-symmetric condition.

Final Logic: Diagonal element equals its own negative only when zero.

🧠 Memory Trick

"Skew diagonal vanishes."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Properties of Skew-Symmetric Matrices

19 If A is an identity matrix I of order 2 × 2, what is the resulting matrix of the composition A² (which is I × I)?

A A Zero matrix

B The Identity matrix I

C A scalar matrix with diagonal 2

D A skew-symmetric matrix

💡 Short Explanation

�� Identity multiplied by itself remains identity. �� Identity behaves like multiplicative 1. �� I^2=I.

✅ Correct Answer Explanation

Identity matrix property: I×I=I Thus: A^2=I^2=I Hence the resulting matrix remains identity matrix.

❓ Why Other Options Are Incorrect

�� Option A → Identity multiplication never gives zero matrix.
�� Option C → Diagonal entries remain 1, not 2.
�� Option D → Identity matrix is symmetric, not skew-symmetric.

🎯 Strategy

Used: Elimination

Application: Remove options contradicting identity multiplication property.

Final Logic: Identity matrix remains unchanged under self-multiplication.

🧠 Memory Trick

"One-like identity stays same."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Identity Matrix Multiplication

20 For a matrix A, the sum representation 1/2 (A + A') + 1/2 (A - A') relies fundamentally on which matrix property to cleanly reconstruct matrix A?

A The cancellation of the transpose matrices (1/2 A' - 1/2 A').

B The fact that all diagonal elements are 0.

C The non-commutativity of addition.

D The multiplication compatibility rule.

💡 Short Explanation

�� Matrix decomposes into symmetric and skew-symmetric parts. �� Opposite transpose terms cancel. �� Remaining terms reconstruct A.

✅ Correct Answer Explanation

Expand the expression: 1/2(A+A^')+1/2(A-A^')=1/2A+1/2A^'+1/2A-1/2A^' The transpose terms cancel: 1/2A^'-1/2A^'=0 leaving: A Thus, the reconstruction depends fundamentally on cancellation of transpose terms.

❓ Why Other Options Are Incorrect

�� Option B → Zero diagonal property is unrelated here.
�� Option C → Matrix addition is commutative, not non-commutative.
�� Option D → Multiplication compatibility is not involved.

🎯 Strategy

Used: Elimination

Application: Expand expression algebraically and identify cancelling terms.

Final Logic: Opposite transpose terms cancel, reconstructing A exactly.

🧠 Memory Trick

"Plus-minus transpose cancels."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Decomposition of Matrices

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