Matrices — Chapter Test | Class 12 IMO

Q1

The transpose of a matrix A, denoted by A', is obtained by:

The transpose of a matrix is formed by turning all its rows into columns and all its columns into rows.

Q2

If A = [[cos θ, −sin θ], [sin θ, cos θ]], then A + A' equals:

A' = [[cos θ, sin θ], [−sin θ, cos θ]]. A + A' = [[cos θ+cos θ, −sin θ+sin θ], [sin θ+(−sin θ), cos θ+cos θ]] = [[2cos θ, 0], [0, 2cos θ]].

Q3

If A = [[1, 2], [3, 4]] and B = [[1, 1], [1, 1]], then (AB)' equals:

AB = [[1×1+2×1, 1×1+2×1], [3×1+4×1, 3×1+4×1]] = [[3, 3], [7, 7]]. Transpose of AB is [[3, 7], [3, 7]].

Q4

If A = [[1, 3], [2, 4]] and B = [[−1, 2], [0, 1]], then the element c₂₁ of the matrix C = AB is:

C = AB = [[1×(−1)+3×0, 1×2+3×1], [2×(−1)+4×0, 2×2+4×1]] = [[−1, 5], [−2, 8]]. c₂₁ is the element in row 2, column 1, which is −2.

Q5

A Delhi Transport Corporation (DTC) depot maintains a fleet of buses. Matrix A represents the number of buses on 3 routes, and Matrix B represents the number of trips per bus on each route. To find total trips on all routes combined, we must:

To calculate the total trips, we multiply the quantity matrix (buses per route) with the rate matrix (trips per route) via matrix multiplication.

Q6

If A is a 2×3 matrix and B is a 3×2 matrix, then the order of BA is:

B is 3×2 and A is 2×3. The product BA is defined because columns of B (2) equals rows of A (2). The order is (rows of B) × (columns of A) = 3×3.

Q7

If matrix A = [[1, 1, 1], [1, 1, 1], [1, 1, 1]], then Aⁿ (where n is a positive integer) is equal to:

A² = A × A = [[3, 3, 3], [3, 3, 3], [3, 3, 3]] = 3A. A³ = A² × A = 3A × A = 3A² = 3(3A) = 9A = 3²A. By induction, Aⁿ = 3^(n−1) A.

Q8

In a town, the number of males and females in three localities are represented by a matrix M = [[200, 180], [150, 160], [100, 120]] where rows represent localities and columns represent males and females. The total number of people in locality 2 is:

Locality 2 corresponds to row 2: [150, 160]. Total = 150 + 160 = 310.

Q9

The transpose of a row matrix is always a:

Transpose of an m×n matrix is n×m. A row matrix is 1×n, so its transpose is n×1, which is a column matrix.

Q10

If A and B are matrices of same order, under what condition is (A + B)² = A² + 2AB + B² true?

Expanding (A + B)² gives (A + B)(A + B) = A² + AB + BA + B². For this to equal A² + 2AB + B², we must have AB = BA (the matrices commute).

Q11

A dietician wishes to mix two types of foods. Matrix M represents the vitamin content per unit of food. Vector X represents the units of food consumed. The product MX gives:

Multiplying the vitamin content matrix (M) by the consumption vector (X) mathematically yields a new vector representing the total intake of each specific vitamin.

Q12

Let A be a 2 × 2 matrix such that trace(A) = 5 and A² - 5A + 6I = O. What are the possible values for the elements on the principal diagonal of an equivalent diagonal matrix?

The characteristic equation of the matrix is given by x² - 5x + 6 = 0, whose roots (eigenvalues) are 2 and 3. The sum of the roots is 5, which matches the trace. The diagonal entries of the equivalent diagonal matrix are its eigenvalues, 2 and 3.

Q13

If A is an involuntary matrix, which condition holds true?

An involutory matrix is a matrix that is its own inverse, meaning that when multiplied by itself, it yields the identity matrix, A² = I.

Q14

If a matrix M = [[1, 2], [2, 1]] and N is a matrix such that M + N is a zero matrix, then N is:

M + N = 0 implies N = −M = [[−1, −2], [−2, −1]].

Q15

A scalar matrix is a diagonal matrix in which all the principal diagonal elements are:

A scalar matrix is a special type of diagonal matrix where all diagonal entries are equal to a constant scalar.