CUET UG Applied Mathematics Booster Test 3 - Matrix Fundamentals and Types -

🚀

CUET UG Applied Mathematics Booster Test 3 - Matrix Fundamentals and Types

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 3 - Matrix Fundamentals and Types

1 / 20

QUESTION 1 OF 20

Let A be an arrangement of complex numbers z = x + iy. Does a matrix strictly require its elements to be real variables or functions?

Answer Review

1 Let A be an arrangement of complex numbers z = x + iy. Does a matrix strictly require its elements to be real variables or functions?

A Yes, matrices only handle real functions.

B No, the elements can be complex numbers or functions.

C Yes, matrices cannot represent algebraic expressions.

D No, but complex numbers are only allowed in diagonal matrices.

💡 Short Explanation

�� Matrix elements may be real, complex, or algebraic expressions. �� A matrix is simply an ordered rectangular arrangement. �� Complex numbers like 2+i are valid entries.

✅ Correct Answer Explanation

A matrix is defined as a rectangular array of elements arranged in rows and columns. These elements are not restricted to only real numbers. They may be complex numbers, variables, algebraic expressions, or functions depending on the mathematical context. → Option B is correct because complex numbers such as z=x+iy are perfectly acceptable as matrix entries. → Option A is incorrect because matrices are not limited to real-valued entries. → Option C is incorrect because algebraic expressions are also commonly used inside matrices. → Option D is incorrect because complex numbers can occur anywhere in the matrix, not only along the diagonal.

❓ Why Other Options Are Incorrect

�� Option A → Matrices are not restricted to real functions only. Complex entries are valid.
�� Option C → Matrices can contain algebraic expressions and symbolic quantities.
�� Option D → Complex numbers may appear in any position of a matrix, not just diagonal positions.

🎯 Strategy

Used: Elimination

Application: Eliminate statements imposing unnecessary restrictions on matrix elements. Matrix theory allows broad classes of entries.

Final Logic: Since matrices may contain complex numbers and functions, Option B is the only universally correct statement.

🧠 Memory Trick

"Matrix entries = Any Math Objects."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Definition of Matrix and Elements

2 Given a matrix C = [cᵢⱼ] where cᵢⱼ = |i - j| / 3. Match the calculated elements:

List I	List II
1. c₁₁	a. 2/3
2. c₁₂	b. 1/3
3. c₃₁	c. 0
4. c₂₃	d. 1/3

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

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

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

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

💡 Short Explanation

�� Use formula c_(ij)=∣i-j∣/3. �� Substitute row and column indices directly. �� Match computed values carefully.

✅ Correct Answer Explanation

Compute each term: c_(11)=∣1-1∣/3=0c_(12)=∣1-2∣/3=1/3c_(31)=∣3-1∣/3=2/3c_(23)=∣2-3∣/3=1/3 Thus: c_(11)→0→ (c) c_(12)→1/3→ (b) c_(31)→2/3→ (a) c_(23)→1/3→ (d) Hence Option A is correct.

❓ Why Other Options Are Incorrect

�� Option A → Mismatches computed substitutions.
�� Option B → Incorrectly assigns c_(11)≠2/3.
�� Option D → Incorrect value mapping for all major entries.

🎯 Strategy

Used: Substitution

Application: Substitute index values directly into the matrix formula.

Final Logic: Direct evaluation of each element uniquely matches Option A.

🧠 Memory Trick

"Put i, j → Compute → Match."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Construction of Matrices from Formulae

3 If a matrix has an odd number of total elements E greater than 2, which of the following MUST be true regarding its order m × n?
1. m and n must both be odd integers.
2. The matrix cannot be of order 2 × 3.
3. If E is a prime number, the matrix must be of order 1 × p or p × 1.
4. The order of a matrix can be a fraction.

A 1, 2, 4

B 1, 2

C 1, 2, 3

D 1, 2, 3, 4

💡 Short Explanation

�� Total elements = Number of rows × Number of columns. �� An odd product requires both factors to be odd. �� Matrix orders are always whole numbers.

✅ Correct Answer Explanation

If a matrix has m rows and n columns, then the total number of elements is: E = m × n → Since E is an odd number, both m and n must be odd integers. If either m or n were even, their product would be even. Therefore, Statement 1 is correct. → A matrix of order 2 × 3 has: 2 × 3 = 6 elements, which is an even number. Hence, such a matrix cannot have an odd number of elements. Therefore, Statement 2 is correct. → If E is a prime number p, its only positive factors are 1 and p. Therefore, the only possible matrix orders are: 1 × p or p × 1 Hence, Statement 3 is correct. → Statement 4 is incorrect because the order of a matrix represents the number of rows and columns, which must always be positive integers. Fractional matrix orders are not defined. → Therefore, Statements 1, 2, and 3 are correct.

❓ Why Other Options Are Incorrect

�� Option A → Incorrect because Statement 3 is also true when the number of elements is prime.
�� Option B → Incorrect because it omits Statement 3.
�� Option D → Incorrect because Statement 4 is mathematically invalid.

🎯 Strategy

Used

Elimination

Application:

�� Use the properties of odd numbers and prime numbers to test each statement and eliminate the option containing the false statement.

Final Logic:

�� An odd number of elements requires odd dimensions, and a prime number of elements allows only the orders 1 × p or p × 1.

🧠 Memory Trick

"Odd × Odd = Odd; Prime = 1 × p."

📚 Class XII – NCERT Mathematics Part I – Chapter: Matrices – Topic: Order of a Matrix and Elements of a Matrix

4 Which is an INCORRECT interpretation of the general element notation aᵢⱼ = 0, ∀ 1 ≤ i ≤ m, 1 ≤ j ≤ n, ∀ i, j ∈ N?

A This defines a zero matrix of order m × n.

B It implies every element in the matrix is null.

C i and j can easily take fractional values to represent positions.

D The notation comprehensively maps all positions in the rectangular array.

💡 Short Explanation

�� Matrix indices are natural numbers. �� Fractional positions are invalid. �� Zero matrix means all entries are zero.

✅ Correct Answer Explanation

In matrix notation, indices i and j represent row and column positions. These indices belong to natural numbers only. → Option C is incorrect because fractional indices like 1.5 are not allowed in matrix positioning. → Option A is correct because all entries being zero defines a zero matrix. → Option B is correct because every element equals zero. → Option D is correct because the notation spans all row-column positions.

❓ Why Other Options Are Incorrect

�� Option A → Correctly defines a zero matrix.
�� Option B → "Null" here means all entries are zero, which is valid.
�� Option D → The notation indeed represents all positions in the matrix.

🎯 Strategy

Used: Extreme Word Filter

Application: Watch for mathematically impossible claims like "fractional positions."

Final Logic: Matrix coordinates must be natural numbers, making Option C incorrect.

🧠 Memory Trick

"Rows & Columns count whole numbers only."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: General Form of Matrix

5 A manufacturer mixes different elements into matrices. A matrix is created where the condition m ≠ n holds. Which operation intrinsically requires changing its rectangular form to a square matrix to be valid?

A Scalar multiplication

B Transpose

C Finding a determinant value

D Matrix addition

💡 Short Explanation

�� Determinants exist only for square matrices. �� Rectangular matrices cannot have determinants. �� Other listed operations work for rectangular matrices.

✅ Correct Answer Explanation

Determinants are defined only for square matrices where the number of rows equals the number of columns. → Since m≠n, the matrix is rectangular, so determinant calculation is invalid. → Scalar multiplication works for any matrix. → Transpose works for every matrix. → Matrix addition works if matrices have the same order. Therefore Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Scalar multiplication is valid for all matrix orders.
�� Option B → Every matrix possesses a transpose.
�� Option D → Addition depends only on equal order, not squareness.

🎯 Strategy

Used: Elimination

Application: Identify which operation has a strict square-matrix condition.

Final Logic: Determinants alone require m=n.

🧠 Memory Trick

"Determinant Demands Square."

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

6 For a constraint region defining the full main diagonal of a matrix, why must the matrix strictly be a square matrix?

A Because rows and columns must be infinite.

B Because diagonal elements aᵢⱼ where i = j do not align from top-left to bottom-right evenly in rectangular matrices.

C Because scalar multiplication only works on square matrices.

D Because m ≠ n defines the diagonal space.

💡 Short Explanation

�� Main diagonal requires equal row-column pairing. �� Square matrices provide complete diagonal alignment. �� Rectangular matrices lack full diagonal symmetry.

✅ Correct Answer Explanation

In a square matrix, each row has a corresponding column, allowing diagonal elements a_(11),a_(22),a_(33),… → In a rectangular matrix, unequal numbers of rows and columns prevent a complete main diagonal extending uniformly. → Hence Option B correctly explains why diagonal structure fundamentally belongs to square matrices.

❓ Why Other Options Are Incorrect

�� Option A → Infinity has no relation to matrix diagonals.
�� Option C → Scalar multiplication works for all matrices.
�� Option D → m≠n defines a rectangular matrix, not diagonal structure.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match the option that directly explains diagonal alignment.

Final Logic: Equal row-column pairing is essential for a full main diagonal.

🧠 Memory Trick

"Diagonal needs equal dimensions."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Diagonal Elements

7 In representing monthly EMI payments sequentially over 'n' periods as a Row Matrix, what will be the resulting matrix dimension?

A n × 1

B 1 × n

C n × n

D 1 × 1

💡 Short Explanation

�� Row matrix has exactly one row. �� Number of columns equals number of entries. �� Hence dimension becomes 1×n.

✅ Correct Answer Explanation

A row matrix contains one row and multiple columns. → If EMI values are listed across n periods horizontally, then: Order=1×n → Option B correctly represents this structure.

❓ Why Other Options Are Incorrect

�� Option A → Represents a column matrix, not a row matrix.
�� Option C → Indicates a square matrix.
�� Option D → Represents only one element.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Identify rows first, then columns.

Final Logic: One row and n entries gives 1×n.

🧠 Memory Trick

"Row → One Row."

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

8 A 3-day moving average is represented as a Column matrix X = [xᵢⱼ]ₘₓₙ. Which specific coordinate condition analytically confirms this structure?

A m = 1, n = 3

B m = 3, n = 3

C m = 3, n = 1

D m = 0, n = 3

💡 Short Explanation

�� Column matrices have one column only. �� Three entries imply three rows. �� Thus order becomes 3×1.

✅ Correct Answer Explanation

A column matrix contains exactly one column. → Since the moving average stores 3 values vertically: m=3,n=1 → Therefore the matrix order is 3×1, making Option C correct.

❓ Why Other Options Are Incorrect

�� Option A → Represents a row matrix.
�� Option B → Represents a square matrix.
�� Option D → A matrix cannot have zero rows.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Count rows and columns separately.

Final Logic: Three vertical entries mean 3 rows and 1 column.

🧠 Memory Trick

"Column → One Column."

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

9 If you randomly select a matrix that acts as the universal additive identity for Aₘₓₙ, what is the probability it is an m × n Zero matrix?

A 0

B 0.5

C 1

D Cannot be determined

💡 Short Explanation

�� Additive identity matrix is always the zero matrix. �� A+0=A. �� Therefore probability is certain.

✅ Correct Answer Explanation

The additive identity for matrices is the zero matrix because: A+O=A for every matrix A. → Hence any matrix serving as universal additive identity must necessarily be a zero matrix. → Therefore probability = 1.

❓ Why Other Options Are Incorrect

�� Option A → Impossible because additive identity always exists as zero matrix.
�� Option B → There is no randomness once additive identity is defined.
�� Option D → The result is completely determined mathematically.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Connect "additive identity" directly with "zero matrix."

Final Logic: Universal additive identity uniquely means zero matrix.

🧠 Memory Trick

"Add Zero, Nothing Changes."

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

10 Two vector matrices V₁ = [x+y 2] and V₂ = [5 y-x] are strictly equal. What are the analytical values of x and y?

A x = 3.5, y = 1.5

B x = 1.5, y = 3.5

C x = 2, y = 3

D x = 5, y = -3

💡 Short Explanation

�� Equal matrices have equal corresponding entries. �� Form simultaneous equations. �� Solve using substitution.

✅ Correct Answer Explanation

Since equal matrices have equal corresponding elements: x+y=5y-x=2 Adding: 2y=7⇒y=3.5 Substitute into first equation: x+3.5=5x=1.5 Thus: x=1.5,y=3.5 Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Reverses values of x and y.
�� Option C → Does not satisfy y-x=2.
�� Option D → Violates both equations simultaneously.

🎯 Strategy

Used: Substitution

Application: Convert equality of matrices into simultaneous equations.

Final Logic: Solving the equations gives x=1.5, y=3.5.

🧠 Memory Trick

"Equal matrices → Equal positions."

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

11 In calculating the area of a triangle using determinants of a 3 × 3 coordinate matrix, do the diagonal elements solely dictate the area?

A Yes, only the diagonal is evaluated.

B No, the determinant expansion involves cross-multiplying both diagonal and non-diagonal elements.

C Yes, non-diagonal elements are always zero.

D No, the area formula does not use determinants.

💡 Short Explanation

�� Determinants use all relevant matrix elements. �� Both diagonal and non-diagonal entries contribute. �� Triangle area formulas commonly use determinants.

✅ Correct Answer Explanation

The determinant of a 3×3 matrix is calculated using products involving both diagonal and non-diagonal elements. → In coordinate geometry, the area of a triangle is often determined using determinant expansion. → Therefore, diagonal elements alone do not determine the area. → Option B correctly explains that determinant evaluation requires cross-multiplication involving multiple matrix positions.

❓ Why Other Options Are Incorrect

�� Option A → Determinants are not evaluated using only diagonal elements.
�� Option C → Non-diagonal elements in coordinate matrices are generally not zero.
�� Option D → The determinant method is a standard formula for triangle area calculation.

🎯 Strategy

Used: Elimination

Application: Remove options contradicting determinant expansion rules from matrix algebra.

Final Logic: Determinants involve all important matrix entries, not only diagonals.

🧠 Memory Trick

"Determinant uses full matrix, not just diagonal."

📚 Class XII – NCERT Mathematics – Chapter: Determinants – Topic: Area of Triangle Using Determinants

12 If the integral value of every non-diagonal element in a square matrix A is logically set to 0, and aᵢᵢ = i, what is the exact element a₂₃?

A 2

B 3

C 0

D 5

💡 Short Explanation

�� Non-diagonal elements satisfy i≠j. �� Given all non-diagonal entries are zero. �� Here 2≠3.

✅ Correct Answer Explanation

The matrix condition states that every non-diagonal element is zero. → Element a_(23)lies in row 2 and column 3, so 2≠3. → Hence a_(23)is a non-diagonal element and must equal 0.

❓ Why Other Options Are Incorrect

�� Option A → 2 applies only to diagonal condition a_(ii)=i.
�� Option B → Incorrect because a_(23)is not a diagonal element.
�� Option D → No rule produces value 5 here.

🎯 Strategy

Used: Substitution

Application: Check whether row index equals column index.

Final Logic: Since 2≠3, the element is non-diagonal and equals zero.

🧠 Memory Trick

"If i≠j, entry becomes 0."

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

13 Which analytical condition distinguishes a general diagonal matrix from a scalar matrix?

A A diagonal matrix has all non-diagonal elements as 1.

B A diagonal matrix can have varying non-zero diagonal elements, while a scalar matrix must have identical diagonal elements.

C A scalar matrix is rectangular.

D A diagonal matrix must have its diagonal elements equal to 1.

💡 Short Explanation

�� Scalar matrices are special diagonal matrices. �� Diagonal entries in scalar matrices are equal. �� General diagonal matrices may have different diagonal values.

✅ Correct Answer Explanation

In a diagonal matrix, only non-diagonal entries are required to be zero. The diagonal entries may differ. → In a scalar matrix, all diagonal entries must be equal to the same constant k. → Hence Option B correctly distinguishes the two matrix types.

❓ Why Other Options Are Incorrect

�� Option A → Non-diagonal elements in diagonal matrices are zero, not 1.
�� Option C → Scalar matrices are always square matrices.
�� Option D → Diagonal matrices need not have diagonal entries equal to 1; that condition defines identity matrices.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Compare definitions of diagonal, scalar, and identity matrices carefully.

Final Logic: Equality of diagonal entries is the defining extra condition for scalar matrices.

🧠 Memory Trick

"Scalar = Same diagonal values."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Scalar and Diagonal Matrices

14 If matrix A is a scalar matrix of order 3 × 3 with diagonal element k, what is the result of pulling the scalar factor k out of the matrix?

A A Zero Matrix

B A Rectangular Matrix

C An Identity matrix of order 3 (I₃)

D A Column matrix

💡 Short Explanation

�� Scalar matrix = kI_n. �� Factoring out k leaves identity matrix. �� Identity matrix has 1s on diagonal.

✅ Correct Answer Explanation

A scalar matrix of order 3×3 can be written as: A=kI_3 → Factoring out k leaves the identity matrix I_3. → Therefore the remaining structure is an identity matrix of order 3.

❓ Why Other Options Are Incorrect

�� Option A → The matrix is not necessarily zero unless k=0.
�� Option B → Scalar matrices are square, not rectangular.
�� Option D → Identity matrices are not column matrices.

🎯 Strategy

Used: Substitution

Application: Rewrite scalar matrix using standard algebraic representation.

Final Logic: Scalar matrices are scalar multiples of identity matrices.

🧠 Memory Trick

"Scalar matrix = constant × identity."

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

15 If A is any square matrix and I is the identity matrix of the same order, which multiplicative property holds true?

A AI = IA = A

B AI = I

C AI = 0

D A + I = 0

💡 Short Explanation

�� Identity matrix acts like multiplicative identity. �� Multiplying by identity leaves matrix unchanged. �� Order compatibility is necessary.

✅ Correct Answer Explanation

The identity matrix behaves like the number 1 in ordinary multiplication. → For any square matrix A: AI=IA=A → Thus multiplying on either side by the identity matrix leaves the matrix unchanged.

❓ Why Other Options Are Incorrect

�� Option B → Product equals A, not identity matrix.
�� Option C → Multiplication with identity does not give zero matrix.
�� Option D → Addition with identity matrix does not generally produce zero.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Use the analogy between the identity matrix and the multiplicative identity, 1.

Final Logic: Identity multiplication preserves the original matrix.

🧠 Memory Trick

"Identity changes nothing."

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

16 In the context of the matrix equation A = Iₙ A, why is the specific notation Iₙ used instead of just I?

A To indicate that it is an inverse matrix.

B To emphasize that the identity matrix must match the specific square order 'n' of matrix A for defined multiplication.

C To denote a column matrix of size n.

D To represent imaginary numbers.

💡 Short Explanation

�� Matrix multiplication requires compatible orders. �� I_n specifies the order n×n. �� Identity matrix must match matrix A.

✅ Correct Answer Explanation

Matrix multiplication is defined only when dimensions are compatible. → If A is of order n×n, then the identity matrix used must also be of order n×n. → Therefore the notation I_n specifies the required order explicitly.

❓ Why Other Options Are Incorrect

�� Option A → I_n does not denote inverse matrices.
�� Option C → Identity matrices are square matrices, not column matrices.
�� Option D → The notation has no relation to imaginary numbers.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Check matrix multiplication compatibility conditions.

Final Logic: Identity matrix order must match the square matrix order.

🧠 Memory Trick

"Same size for multiplication."

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

17 If a matrix has 6 elements, which of the following is NOT a structurally possible order for the matrix?

A 1 × 6

B 2 × 3

C 3 × 2

D 4 × 2

💡 Short Explanation

�� Total elements = rows × columns. �� Valid orders must multiply to 6. �� 4×2=8.

✅ Correct Answer Explanation

Number of elements in a matrix equals: Total elements=m×n → Orders 1×6, 2×3, and 3×2 all contain 6 elements. → But 4×2=8, so this order is impossible for a 6-element matrix.

❓ Why Other Options Are Incorrect

�� Option A → Contains exactly 6 elements.
�� Option B → Valid because 2×3=6.
�� Option C → Valid because 3×2=6.

🎯 Strategy

Used: Substitution

Application: Multiply rows and columns for each option.

Final Logic: Only 4×2 does not produce 6 elements.

🧠 Memory Trick

"Rows × columns = total entries."

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

18 Construct matrix A = [aᵢⱼ] of order 3 × 3 where aᵢⱼ = |i - j|. What specific type of matrix is this analytically?

A Identity Matrix

B Scalar Matrix

C Symmetric Matrix (since |i - j| = |j - i|)

D Skew-Symmetric Matrix

💡 Short Explanation

�� Absolute value satisfies symmetry property. �� a_(ij)=a_(ji). �� Symmetric matrices satisfy A^T=A.

✅ Correct Answer Explanation

Given: a_(ij)=∣i-j∣=∣j-i∣=a_(ji) → Since corresponding elements across the diagonal are equal, the matrix satisfies the condition for symmetry. → Therefore the matrix is symmetric.

❓ Why Other Options Are Incorrect

�� Option A → Identity matrices require 1s on diagonal and 0 elsewhere.
�� Option B → Scalar matrices require equal diagonal entries and zero non-diagonal entries.
�� Option D → Skew-symmetric matrices satisfy a_(ij)=-a_(ji), which is not true here.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Compare the given relation with standard matrix definitions.

Final Logic: Equality a_(ij)=a_(ji)confirms symmetry.

🧠 Memory Trick

"Mirror equality ⇒ Symmetric."

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

📄 Passage

Leontief input-output models comprehensively analyze the economy using advanced matrix operations. An economy's viability heavily depends on various sectors, such as the automobile and electrical sectors, effectively sharing resources. If the input-output coefficient matrix is denoted by A, and the identity matrix of the corresponding same order is I, the system's viability is rigorously tested using the Hawkins-Simon conditions. These conditions dictate that the determinant |I - A| must be positive, and additionally, the diagonal elements of (I - A) must also be positive. Through tabular data matrices, we accurately compute the specific requirements needed to fulfill market demands. The required output is ultimately found using the inverse equation X = (I - A)⁻¹ D. Here, D represents the constant demand matrix structured in a specific coordinate-like column format.

Question:
In the Leontief model passage, if the coefficient tabular data is represented by matrix A, what operational role does the expression (I - A) serve?

A It proves the matrix is a zero matrix.

B It generates the diagonal elements needed to verify the Hawkins-Simon conditions for economic viability.

C It converts a square matrix into a row matrix.

D It directly acts as the demand matrix D.

💡 Short Explanation

�� Hawkins-Simon conditions use I-A. �� Positivity of determinant and diagonal entries is checked. �� Used in economic viability testing.

✅ Correct Answer Explanation

In the Leontief input-output model, economic viability is tested using the matrix I-A. → Hawkins-Simon conditions require: • determinant ∣I-A∣>0 • positive diagonal elements of I-A → Thus I-A plays a key role in checking economic feasibility conditions.

❓ Why Other Options Are Incorrect

�� Option A → I-A is not necessarily a zero matrix.
�� Option C → No row-matrix conversion occurs.
�� Option D → Demand matrix is separately represented by D.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match the passage statement with Leontief model terminology.

Final Logic: I-A is specifically used in Hawkins-Simon viability conditions.

🧠 Memory Trick

"Viability check uses I-A."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Applications of Matrices in Economics

📄 Passage

Question:
According to the passage, the demand matrix D is typically structured in what format to accurately allow calculation of the required output X = (I - A)⁻¹ D?

A 1 × 1 scalar

B Square matrix matching order of A

C Column matrix acting as a coordinate demand vector

D Row matrix exclusively

💡 Short Explanation

�� Demand values are arranged vertically. �� Output equations use column vectors. �� Compatible multiplication with inverse matrix is required.

✅ Correct Answer Explanation

In the Leontief model, the demand matrix D represents sector-wise demand values. → These values are arranged in column-vector form so that multiplication with (I-A)^(-1)becomes dimensionally valid. → Therefore D acts as a coordinate-like demand column matrix.

❓ Why Other Options Are Incorrect

�� Option A → Demand involves multiple sector values, not a single scalar.
�� Option B → Demand matrix need not be square.
�� Option D → A row matrix would generally violate multiplication compatibility.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Verify compatible matrix multiplication structure in the equation.

Final Logic: Demand data must be arranged as a column vector for valid multiplication.

🧠 Memory Trick

"Demand flows downward ⇒ column vector."

📚 Class XII – NCERT Mathematics – Chapter: Matrices – Topic: Input-Output Model

CUET UG Applied Mathematics Booster Test 3 - Matrix Fundamentals and Types

Test Complete!

Answer Review