CUET UG Applied Mathematics Booster Test 3 - Matrix Algebra and Transformations -

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CUET UG Applied Mathematics Booster Test 3 - 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 3 - Matrix Algebra and Transformations

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QUESTION 1 OF 20

If the integral transformation of a matrix X yields kX = [-6 9 ; 12 -3] and the scalar k = -3, what was the original matrix X?

Answer Review

1 If the integral transformation of a matrix X yields kX = [-6 9 ; 12 -3] and the scalar k = -3, what was the original matrix X?

A [2 -3 ; -4 1]

B [-2 3 ; 4 -1]

C [18 -27 ; -36 9]

D [3 -6 ; -9 0]

💡 Short Explanation

�� Original matrix is obtained by dividing each element by k. �� Here k = -3. �� Scalar multiplication preserves matrix order.

✅ Correct Answer Explanation

Given: -3X=[[-6, 9; 12, -3]] To find X, divide every element by -3: X=[[-6/-3, 9/-3; 12/-3, -3/-3]]=[[2, -3; -4, 1]] Thus Option A is correct. Option B changes signs incorrectly. Option C multiplies instead of dividing. Option D does not satisfy scalar multiplication.

❓ Why Other Options Are Incorrect

�� Option B → Signs are reversed incorrectly after division.
�� Option C → Elements are multiplied by 3 instead of divided by -3.
�� Option D → Matrix does not produce the given matrix after multiplication by -3.

🎯 Strategy

Used: Substitution

Application: Multiply each option by -3 and verify which reproduces the given matrix.

Final Logic: Only Option A satisfies (-3)X=given matrix.

🧠 Memory Trick

"Undo scalar multiplication by division."

📚 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 produces zero matrix.

✅ Correct Answer Explanation

Using distributive property: (p+q)A=(2-2)A=0A Multiplying any matrix by zero gives the zero matrix: 0A=O Hence Option C is correct. Option A incorrectly adds magnitudes. Option B ignores scalar effect. Option D incorrectly assumes negative scaling.

❓ Why Other Options Are Incorrect

�� Option A → 2+(-2)≠4.
�� Option B → Scalar becomes zero, not one.
�� Option D → No negative factor remains after addition.

🎯 Strategy

Used: Substitution

Application: Directly substitute p and q into the expression.

Final Logic: Zero scalar times any matrix gives zero matrix.

🧠 Memory Trick

"Zero scalar → Zero matrix."

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

3 Given the matrices
X = [[a, b; c, d]]
and
Y = [[p, q; r, s]],
what is the element at the second row, first column in the sum matrix X + Y?

A a + p

B b + q

C c + r

D d + s

💡 Short Explanation

�� Matrix addition is performed element-wise. �� Corresponding elements are added. �� The second row, first column contains c and r.

✅ Correct Answer Explanation

Two matrices of the same order are added by adding their corresponding elements. Given, X = | a b | | c d | and Y = | p q | | r s | Therefore, X + Y = | a + p b + q | | c + r d + s | → The element in the second row and first column is obtained by adding the corresponding elements c and r. Therefore, Second row, first column = c + r. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Represents the element in the first row, first column.
�� Option B → Represents the element in the first row, second column.
�� Option D → Represents the element in the second row, second column.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

�� Identify the required position in both matrices and add the corresponding elements.

Final Logic:

�� Matrix addition is performed position-wise; therefore, the element in the second row and first column is c + r.

🧠 Memory Trick

"Same Position → Same Addition."

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

4 If matrix addition is associative, i.e., X + (Y + Z) = (X + Y) + Z, what is the probability that changing the parenthesis mathematically alters the final matrix order, assuming X, Y, and Z are all of order 3 × 4?

A 1

B 0.5

C 0

D 0.75

💡 Short Explanation

�� Associativity does not change order. �� All matrices are 3×4. �� Final order remains same.

✅ Correct Answer Explanation

Matrix addition is associative: X+(Y+Z)=(X+Y)+Z Since all matrices are of order 3×4, the resulting matrix also remains 3×4 regardless of grouping. Thus the probability that order changes is: Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Order never changes.
�� Option B → No partial probability exists here.
�� Option D → Associativity fully preserves order.

🎯 Strategy

Used: Elimination

Application: Eliminate all options suggesting order variation.

Final Logic: Associative law preserves matrix order completely.

🧠 Memory Trick

"Brackets move, order stays."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Associative Property of Matrix Addition

5 A moving average matrix D is derived from D = A - B. If a_ij and b_ij are the corresponding elements, what defines the general element d_ij mathematically?

A a_ij + b_ij

B a_ij × b_ij

C a_ij - b_ij

D b_ij - a_ij

💡 Short Explanation

�� Matrix subtraction is element-wise. �� Corresponding entries are subtracted. �� Order must remain same.

✅ Correct Answer Explanation

If: D=A-B then each element of D is: d_(ij)=a_(ij)-b_(ij) Thus Option C is correct. Option A represents addition. Option B represents multiplication. Option D reverses subtraction order.

❓ Why Other Options Are Incorrect

�� Option A → Gives matrix addition.
�� Option B → Matrix subtraction is not multiplication.
�� Option D → Wrong subtraction sequence.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Use the definition of subtraction directly.

Final Logic: Subtract corresponding entries position-wise.

🧠 Memory Trick

"Subtract same-position entries."

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

6 If the EMI matrix E represents a debt, its additive inverse -E represents the equivalent credit. Which algebraic equation correctly defines this relationship using the zero matrix O?

A E - (-E) = O

B E + (-E) = O

C E × (-E) = I

D E + O = -E

💡 Short Explanation

�� Additive inverse cancels matrix. �� Sum becomes zero matrix. �� Similar to real numbers.

✅ Correct Answer Explanation

The additive inverse of a matrix E is: -E By definition: E+(-E)=O where O is the zero matrix. Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → E-(-E)=E+E=2E, not O.
�� Option C → Matrix multiplication does not define additive inverse.
�� Option D → E+O=E, not -E.

🎯 Strategy

Used: Elimination

Application: Recall additive identity property.

Final Logic: A matrix plus its negative equals zero matrix.

🧠 Memory Trick

"Positive plus negative cancels."

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

7 For finding the matrix product AB = P = [p_ik] within a strictly defined constraint region, how is the specific element p_ik calculated?

A Multiplying corresponding elements: a_ik × b_ik

B Sum of the products of elements of the i-th row of A and the k-th column of B

C Sum of the products of elements of the i-th column of A and the k-th row of B

D Adding the i-th row of A to the k-th column of B

💡 Short Explanation

�� Matrix multiplication uses row-column rule. �� Multiply corresponding row and column terms. �� Add obtained products.

✅ Correct Answer Explanation

For matrix multiplication: P=AB The element p_(ik)is obtained by: p_(ik)=∑a_(ij)b_(jk) This means: → Take the i-th row of A → Take the k-th column of B → Multiply corresponding entries → Add them Thus Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Matrix multiplication is not element-wise multiplication.
�� Option C → Uses incorrect row-column pairing.
�� Option D → Addition alone cannot form product entries.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recall standard row-column multiplication rule.

Final Logic: Each product entry comes from row-column multiplication.

🧠 Memory Trick

"Row × Column = Product Entry."

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

8 A complex mixture requires composing matrix A (order m × n) and matrix B (order k × l). Both products AB and BA are viable and defined. What does this algebraically imply about their orders?

A m = n and k = l

B n = k and l = m

C n = l and m = k

D m = 1 and l = 1

💡 Short Explanation

�� For AB: columns of A = rows of B. �� For BA: columns of B = rows of A. �� Hence n=k and l=m.

✅ Correct Answer Explanation

For product AB to exist: n=k For product BA to exist: l=m Therefore both conditions together imply: n=kandl=m Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Square matrices are not necessary.
�� Option C → Gives unrelated equalities.
�� Option D → No such restriction exists.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Check multiplication compatibility conditions carefully.

Final Logic: Inner dimensions must match for both products.

🧠 Memory Trick

"Inside dimensions must agree."

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

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 = BA + CA is always true.

D A(B +

💡 Short Explanation

�� Matrix multiplication is generally non-commutative. �� Correct distributive law keeps A on left side. �� BA and CA are not equivalent to AB and AC.

✅ Correct Answer Explanation

Correct distributive laws are: A(B+C)=AB+AC and (A+B)C=AC+BC Option D incorrectly assumes: AB=BA But matrix multiplication is generally non-commutative. Hence Option D is incorrect.

❓ Why Other Options Are Incorrect

�� Option A → Correct left distributive law.
�� Option B → Correct right distributive law.
�� Option C → Correct subtraction distributive property.

🎯 Strategy

Used: Extreme Word Filter

Application: The word "always" signals possible invalidity in matrix algebra.

Final Logic: Matrix multiplication is not universally commutative.

🧠 Memory Trick

"Matrices don't always commute."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Distributive Properties

10 Which 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 of the same order, but their corresponding elements are unequal.
4. AB = BA only for all rectangular matrices where m ≠ n.

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 definition, order, or numerical values.

✅ Correct Answer Explanation

Matrix multiplication is generally not commutative, that is, AB ≠ BA → Statement 1 is correct because there are situations where AB is defined, but BA is not. Example: If A is of order 2 × 3 and B is of order 3 × 4, then AB is defined (2 × 4), whereas BA is not defined. → Statement 2 is correct because both products may exist 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 are defined 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. Hence, AB ≠ BA in general. → Statement 4 is incorrect because matrix multiplication is not commutative for all rectangular matrices. In fact, for rectangular matrices with (m \ne n), AB and BA are often not even simultaneously defined. Commutativity is a special property that holds only for certain pairs of matrices, not for all rectangular matrices. → 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 correct.
�� Option D → Incorrect because Statement 4 is false.

🎯 Strategy

Used

Option Grouping

Application:

�� Verify each statement independently and eliminate every option containing the false statement about universal commutativity.

Final Logic:

�� Matrix multiplication is non-commutative because the products may differ in existence, order, or corresponding elements.

🧠 Memory Trick

"Defined, Different, Unequal → AB ≠ BA"

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

11 Match the matrix operations to the correct outputs assuming A and B are square matrices of order n, and I is the identity matrix of order n:

List I	List II
1. AI	a. A
2. A(BC)	b. (AB)C
3. IA	c. O
4. O × A (where O is zero matrix)	d. A

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

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

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

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

💡 Short Explanation

�� Identity matrix leaves matrix unchanged. �� Matrix multiplication is associative. �� Zero matrix product gives zero matrix.

✅ Correct Answer Explanation

Using standard matrix properties: AI=A so (1)-(a) A(BC)=(AB)C so (2)-(b) IA=A so (3)-(d) O×A=O so (4)-(c) Hence Option D is correct.

❓ Why Other Options Are Incorrect

�� Option A → Incorrect mapping of zero and associative properties.
�� Option B → Incorrectly matches AI and IA.
�� Option C → Wrongly associates associative property and zero matrix result.

🎯 Strategy

Used: Option Grouping

Application: Use standard matrix identities one by one to match pairs.

Final Logic: Only Option A satisfies all matrix operation properties correctly.

🧠 Memory Trick

"Identity keeps same, zero kills product."

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

12 If A and B are matrices of order 2 × 2 and AB = O, but A ≠ O and B ≠ O, this proves that:

A The matrices must be inverses of each other.

B The zero product property of real numbers does not strictly apply to matrices.

C Matrix A must be a scalar matrix.

D Matrix B must be an identity matrix.

💡 Short Explanation

�� Non-zero matrices can produce zero product. �� Matrix algebra differs from real numbers. �� Zero product law fails generally.

✅ Correct Answer Explanation

In real numbers: ab=0⇒a=0 or b=0 But in matrices: AB=O does not necessarily imply: A=O or B=O Non-zero matrices may multiply to give the zero matrix. Thus, Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Inverse matrices produce identity matrix, not zero matrix.
�� Option C → No condition implies scalar matrix.
�� Option D → Identity matrix cannot produce zero product with non-zero matrix.

🎯 Strategy

Used: Elimination

Application: Compare matrix multiplication rules with real-number properties.

Final Logic: Matrices do not always satisfy zero product law.

🧠 Memory Trick

"Non-zero matrices can vanish on multiplication."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Zero Product in Matrices

13 If matrix A has order m × n, and B has order n × p, then the order of the transposed product matrix (AB)' is:

A m × p

B p × m

C n × n

D p × p

💡 Short Explanation

�� Product AB has order m×p. �� Transpose interchanges rows and columns. �� Final order becomes p×m.

✅ Correct Answer Explanation

Given: A_(m×n),B_(n×p) Then: AB has order m×p Taking transpose: (AB)^' has order p, m Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Represents order of AB before transpose.
�� Option C → No such square order relation exists.
�� Option D → Incorrect transpose dimensions.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Track matrix dimensions step-by-step.

Final Logic: Transpose reverses dimensions of product matrix.

🧠 Memory Trick

"Transpose flips dimensions."

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

14 Which expansion is mathematically accurate for the transpose of a sum, (A + B)'?

A A' × B'

B A' - B'

C A' + B'

D A + B

💡 Short Explanation

�� Transpose distributes over addition. �� Addition property is preserved. �� Standard transpose law applies.

✅ Correct Answer Explanation

Transpose property: (A+B)^'=A^'+B^' Thus Option C is correct. Option A incorrectly multiplies matrices. Option B changes sign unnecessarily. Option D ignores transpose operation.

❓ Why Other Options Are Incorrect

�� Option A → Transpose of sum is not product of transposes.
�� Option B → No subtraction occurs in transpose property.
�� Option D → Transpose operation cannot disappear.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recall standard transpose identities.

Final Logic: Transpose distributes directly across addition.

🧠 Memory Trick

"Transpose respects addition."

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

15 If a matrix P is symmetric, what happens analytically when we take the transpose of its transpose, (P')'?

A It becomes a skew-symmetric matrix.

B It yields the original symmetric matrix P.

C It becomes an identity matrix.

D It yields the zero matrix.

💡 Short Explanation

�� Double transpose returns original matrix. �� Symmetric matrix satisfies P^'=P. �� Hence (P^')^', P.

✅ Correct Answer Explanation

For any matrix: (A^')^', A Since P is symmetric: P^'=P Therefore: (P^')^', P Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Double transpose does not create skew symmetry.
�� Option C → Identity matrix is unrelated here.
�� Option D → No zero matrix condition exists.

🎯 Strategy

Used: Substitution

Application: Substitute P^'=P into transpose identity.

Final Logic: Double transpose restores original matrix.

🧠 Memory Trick

"Transpose twice, original returns."

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

16 For a square matrix A, if A = 1/2 (A + A'), what classification inherently applies to matrix A?

A It is solely a skew-symmetric matrix.

B It is a zero matrix.

C It is a symmetric matrix.

D It is an identity matrix.

💡 Short Explanation

�� 1/2(A+A^')is symmetric part of A. �� If equal to A itself, then A=A^'. �� Hence A is symmetric.

✅ Correct Answer Explanation

The expression: 1/2(A+A^') always represents the symmetric part of matrix A. If: A=1/2(A+A^') then: A=A^' Thus A satisfies the definition of a symmetric matrix. Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Skew-symmetric matrices satisfy A^'=-A.
�� Option B → No condition implies all entries are zero.
�� Option D → Identity matrix is only one special symmetric matrix.

🎯 Strategy

Used: Elimination

Application: Compare given equation with matrix definitions.

Final Logic: Equality with symmetric component implies symmetry.

🧠 Memory Trick

"A+A^' creates symmetry."

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

17 If matrix M is skew-symmetric, then the mathematical expression M + M' equals:

A 2M

B I

C O (Zero matrix)

D -2M

💡 Short Explanation

�� Skew-symmetric means M^'=-M. �� Add both matrices. �� Terms cancel completely.

✅ Correct Answer Explanation

For skew-symmetric matrix: M^'=-M Therefore: M+M^'=M+(-M)=O Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Would occur if M^'=M.
�� Option B → Identity matrix is unrelated.
�� Option D → Incorrect simplification.

🎯 Strategy

Used: Substitution

Application: Replace M^'with -M.

Final Logic: Opposite matrices sum to zero matrix.

🧠 Memory Trick

"Skew + transpose cancels."

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

18 In evaluating a skew-symmetric matrix Q, why does Q' = -Q force the main diagonal elements to zero?

A Identity properties force it.

B For any element q_ii, q_ii = -q_ii, which solves algebraically to 2q_ii = 0, meaning q_ii = 0.

C Diagonal elements are unrelated to the transpose.

D The zero matrix is the only skew-symmetric matrix.

💡 Short Explanation

�� Skew symmetry gives Q^'=-Q. �� Diagonal entries satisfy q_(ii)=-q_(ii). �� Therefore diagonal entries are zero.

✅ Correct Answer Explanation

For skew-symmetric matrix: Q^'=-Q Diagonal elements remain unchanged under transpose: q_(ii)=-q_(ii) Adding both sides: 2q_(ii)=0 Thus: q_(ii)=0 Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Identity matrix has no role here.
�� Option C → Diagonal entries are directly involved in transpose relation.
�� Option D → Many non-zero skew-symmetric matrices exist.

🎯 Strategy

Used: Substitution

Application: Apply skew-symmetric definition to diagonal elements.

Final Logic: Only zero satisfies q_(ii)=-q_(ii).

🧠 Memory Trick

"Diagonal equals its negative → zero."

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

19 Given A = [2 0 ; 0 2], what is the composition of A³?

A [8 0 ; 0 8]

B [6 0 ; 0 6]

C [4 0 ; 0 4]

D [16 0 ; 0 16]

💡 Short Explanation

�� A=2I. �� A^3=(2I)^3=8I. �� Multiply matrix three times.

✅ Correct Answer Explanation

Given: A=[[2, 0; 0, 2]]=2I Then: A^3=(2I)^3=2^3I=8I Thus: A^3=[[8, 0; 0, 8]] Hence Option A is correct.

❓ Why Other Options Are Incorrect

�� Option B → Represents scalar multiplication by 3, not cube.
�� Option C → Gives A^2, not A^3.
�� Option D → Represents 2^4, not 2^3.

🎯 Strategy

Used: Substitution

Application: Rewrite matrix as scalar multiple of identity matrix.

Final Logic: Cubing scalar 2 gives 8.

🧠 Memory Trick

"2^3=8 on diagonal."

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

20 The ability to express any square matrix A as 1/2 (A + A') + 1/2 (A - A') relies on separating the matrix into which two distinct structural components?

A A diagonal and a scalar matrix

B An identity and a zero matrix

C A symmetric and a skew-symmetric matrix

D A row and a column matrix

💡 Short Explanation

�� 1/2(A+A^')is symmetric part. �� 1/2(A-A^')is skew-symmetric part. �� Their sum reconstructs A.

✅ Correct Answer Explanation

Every square matrix can be decomposed as: A=1/2(A+A^')+1/2(A-A^') Where: 1/2(A+A^') is symmetric, and 1/2(A-A^') is skew-symmetric. Thus Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Diagonal/scalar decomposition is not general.
�� Option B → Identity and zero matrices cannot represent all matrices.
�� Option D → Row-column separation is unrelated.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recognize standard decomposition theorem.

Final Logic: Every square matrix splits into symmetric and skew-symmetric parts.

🧠 Memory Trick

"Plus gives symmetric, minus gives skew."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Decomposition into Symmetric and Skew-Symmetric Parts

CUET UG Applied Mathematics Booster Test 3 - Matrix Algebra and Transformations

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