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## Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.5

Choose the correct or the most suitable answer from the given four alternatives.

Question 1.

If a_{ij} = \(\frac{1}{2}\) (3i – 2j) and A = [a_{ij}]_{2×2} is

Solution:

Question 2.

What must be the matrix X, if 2X + \(\left[\begin{array}{ll}{1} & {2} \\ {3} & {4}\end{array}\right]=\left[\begin{array}{ll}{3} & {8} \\ {7} & {2}\end{array}\right]\) ?

Solution:

Question 3.

Which one of the following is not true about the matrix \(\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {0} & {0} \\ {0} & {0} & {5}\end{array}\right]\)?

(a) a scalar matrix

(b) a diagonal matrix

(c) an upper triangular matrix

(d) A lower triangular matrix

Solution:

(b) a diagonal matrix

Question 4.

If A and B are two matrices such that A + B and AB are both defined, then …………

(a) A and B are two matrices not necessarily of same order.

(b) A and B are square matrices of same order.

(c) Number of columns of a is equal to the number of rows of B.

(d) A = B.

Solution:

(b) A and B are square matrices of same order.

Question 5.

If A = \(\left[\begin{array}{rr}{\lambda} & {1} \\ {-1} & {-\lambda}\end{array}\right]\), then for what value of λ, A^{2} = 0?

(a) 0

(b) ±1

(c) -1

(d) 1

Solution:

Question 6.

If and (A + B)^{2} = A^{2} + B^{2}, then the values of a and b are ……………….

(a) a = 4, b = 1

(b) a = 1, b = 4

(c) a = 0, b = 4

(d) a = 2, b = 4

Solution:

Question 7.

If is a matrix satisfying the equation AA^{T} = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to ………….

(a) (2, -1)

(b) (-2, 1)

(c) (2, 1)

(d) (-2, -1)

Solution:

Question 8.

If A is a square matrix, then which of the following is not symmetric?

(a) A + A^{T}

(b) AA^{T}

(c) A^{T}A

(d) A – A^{T}

Solution:

(b) AA^{T}

Question 9.

If A and B are symmetric matrices of order n, where (A ≠ B), then …………….

(a) A + B is skew-symmetric

(b) A + B is symmetric

(c) A + B is a diagonal matrix

(d) A + B is a zero matrix

Solution:

(b) A + B is symmetric

Question 10.

If and if xy = 1, then det (AA^{T}) is equal to …………..

(a) (a – 1)^{2}

(b) (a^{2} + 1)^{2}

(c) a^{2} – 1

(d) (a^{2} – 1)^{2}

Solution:

Question 11.

The value of x, for which the matrix is singular is ………….

(a) 9

(b) 8

(c) 7

(d) 6

Solution:

(b) Hint: Given A is a singular matrix ⇒ |A| = 0

⇒ e^{x-2}.e^{2x+3} – e^{2+x}.e^{7+x} = 0

⇒ e^{3x+1} – e^{9+2x} = 0

⇒ e^{3x+1} = e^{9+2x}

⇒ 3x + 1 = 9 + 2x

⇒ 3x – 2x = 9 – 1

⇒ x = 8

Question 12.

If the points (x, -2), (5, 2), (8, 8) are collinear, then x is equal to …………

(a) -3

(b) \(\frac{1}{3}\)

(c) 1

(d) 3

Solution:

(d) Hint: Given that the points are collinear

So, area of the triangle formed by the points = 0

Question 13.

Solution:

Question 14.

If the square of the matrix is the unit matrix of order 2, then α, β and γ should satisfy the relation.

(a) 1 + α^{2} + βγ = 0

(b) 1 – α^{2} – βγ = 0

(c) 1 – α^{2} + βγ = 0

(d) 1 + α^{2} – βγ = 0

Solution:

Question 15.

(a) Δ

(b) kΔ

(c) 3kΔ

(d) k^{3}Δ

Solution:

Question 16.

A root of the equation is …………….

(a) 6

(b) 3

(c) 0

(d) -6

Solution:

Question 17.

The value of the determinant of is ……………

(a) -2abc

(b) abc

(c) 0

(d) a^{2} + b^{2} + c^{2}

Solution:

Question 18.

If x_{1}, x_{2}, x_{3} as well as y_{1}, y_{2}, y_{3} are in geometric progression with the same common ratio, then the points (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}) are

(a) vertices of an equilateral triangle

(b) vertices of a right-angled triangle

(c) vertices of a right-angled isosceles triangle

(d) collinear

Solution:

(d)

Question 19.

If \(\lfloor.\rfloor\) denotes the greatest integer less than or equal to the real number under consideration and -1 ≤ x < 0, 0 ≤ y < 1, 1 ≤ z ≤ 2, then the value of the determinant is …………..

(a) \(\lfloor z\rfloor\)

(b) \(\lfloor y\rfloor\)

(c) \(\lfloor x\rfloor\)

(d) \(\lfloor x\rfloor+ 1\)

Solution:

(a) Hint: From the given values

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Question 20.

If a ≠ b, b, c satisfy then abc = ……………..

(a) a + b + c

(b) 0

(c) b^{3}

(d) ab + bc

Solution:

(c) Hint: Expanding along R_{1},

a(b^{2} – ac) – 2b (3b – 4c) + 2c (3a – 4b) = 0

(b^{2} – ac) (a – b) = 0

b^{2} = ac (or) a = b

⇒ abc = b(b^{2}) = b^{3}

Question 21.

If then B is given by ………………..

(a) B = 4A

(b) B = -4A

(c) B = -A

(d) B = 6A

Solution:

Question 22.

If A is skew-symmetric of order n and C ¡s a column matrix of order n × 1, then C^{T} AC is ……………..

(a) an identity matrix of order n

(b) an identity matrix of order 1

(e) a zero matrix of order I

(d) an Identity matrix of order 2

Solution:

(c) a zero matrix of order I

Hint: Given A is of order n × n

C is of order n × 1

so, CT is of order 1 × n

Let it be equal to (x) say

Taking transpose on either side

(C^{T}, AC)^{T} (x)^{T} .

(i.e.) C^{T}(A^{T})(C) = x

C^{T}(-A)(C) = x

⇒ C^{T}AC = -x

⇒ x = -x

⇒ 2x = 0

⇒ x = 0

Question 23.

The matrix A satisfying the equation is ……………

Solution:

Question 24.

If A + I = , then (A + I) (A – I) is equal to …………….

Solution:

Question 25.

Let A and B be two symmetric matrices of the same order. Then which one of the following statements is not true?

(a) A + B ¡s a symmetric matrix

(b) AB ¡s a symmetric matrix

(c) AB = (BA)^{T}

(d) A^{T}B = AB^{T}

Solution:

(b) AB ¡s a symmetric matrix