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## Tamilnadu Samacheer Kalvi 11th Business Maths Solutions Chapter 1 Matrices and Determinants Ex 1.5

### Samacheer Kalvi 11th Business Maths Matrices and Determinants Ex 1.5 Text Book Back Questions and Answers

Question 1.
The value of x if $$\left|\begin{array}{lll} 0 & 1 & 0 \\ x & 2 & x \\ 1 & 3 & x \end{array}\right|=0$$ is
(a) 0, -1
(b) 0, 1
(c) -1, 1
(d) -1, -1
(b) 0, 1
Hint:
0 – 1[x2 – x] + 0 = 0
⇒ x2 – x = 0
⇒ x(x – 1) = 0
⇒ x = 0 (or) x = 1

Question 2.
The value of $$\left|\begin{array}{lll} 2 x+y & x & y \\ 2 y+z & y & z \\ 2 z+x & z & x \end{array}\right|$$ is
(a) xyz
(b) x + y + z
(c) 2x + 2y + 2z
(d) 0
(d) 0
Hint:
= $$\left|\begin{array}{lll} 2 x & x & y \\ 2 y & y & z \\ 2 z & z & x \end{array}\right|$$ C1 → C1 – C3
= 0 (C1 and C2 are proportional)

Question 3.
The cofactor of -7 in the determinant $$\left|\begin{array}{rrr} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{array}\right|$$ is
(a) -18
(b) 18
(c) -7
(d) 7
(b) 18
Hint:
A cofactor of -7 = $$\left|\begin{array}{rr} 2 & -3 \\ 6 & 0 \end{array}\right|$$
= 0 + 18
= 18

Question 4.
If Δ = $$\left|\begin{array}{lll} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{array}\right|$$ then $$\left|\begin{array}{lll} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \end{array}\right|$$ is
(a) Δ
(b) -Δ
(c) 3Δ
(d) -3Δ
(b) -Δ
Hint:
$$\left|\begin{array}{lll} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \end{array}\right|=-\left|\begin{array}{lll} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{array}\right|$$ R1 ↔ R2
= -Δ

Question 5.
The value of the determinant $$\left|\begin{array}{lll} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array}\right|^{2}$$ is
(a) abc
(b) 0
(c) a2b2c2
(d) -abc
(c) a2b2c2
Hint:
$$a^{2} b^{2} c^{2}\left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right|$$
= a2b2c2 × 12
= a2b2c2

Question 6.
If A is square matrix of order 3 then |kA| is:
(a) k|A|
(b) -k|A|
(c) k3|A|
(d) -k3|A|
(c) k3|A|

Question 7.

Question 8.
The inverse matrix of $$\left(\begin{array}{cc} \frac{4}{5} & \frac{5}{12} \\ \frac{2}{5} & \frac{1}{2} \end{array}\right)$$ is
(a) $$\frac{7}{30}\left(\begin{array}{cc} \frac{1}{2} & \frac{5}{12} \\ \frac{2}{5} & \frac{4}{5} \end{array}\right)$$
(b) $$\frac{7}{30}\left(\begin{array}{cc} \frac{1}{2} & \frac{-5}{12} \\ \frac{-2}{5} & \frac{1}{5} \end{array}\right)$$
(c) $$\frac{30}{7}\left(\begin{array}{rr} \frac{1}{2} & \frac{5}{12} \\ \frac{2}{5} & \frac{4}{5} \end{array}\right)$$
(d) $$\frac{30}{7}\left(\begin{array}{rr} \frac{1}{2} & \frac{-5}{12} \\ \frac{-2}{5} & \frac{4}{5} \end{array}\right)$$
(c) $$\frac{30}{7}\left(\begin{array}{rr} \frac{1}{2} & \frac{5}{12} \\ \frac{2}{5} & \frac{4}{5} \end{array}\right)$$

Question 9.
If A = $$\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$$ such that ad – bc ≠ 0 then A-1 is:
(a) $$\frac{1}{a d-b c}\left[\begin{array}{cc} d & b \\ -c & a \end{array}\right]$$
(b) $$\frac{1}{a d-b c}\left[\begin{array}{ll} d & b \\ c & a \end{array}\right]$$
(c) $$\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right]$$
(d) $$\frac{1}{a d-b c}\left[\begin{array}{ll} d & -b \\ c & a \end{array}\right]$$
(c) $$\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right]$$
Hint:

Question 10.
The number of Hawkins-Simon conditions for the viability of input-output analysis is:
(a) 1
(b) 3
(c) 4
(d) 2
(d) 2

Question 11.
The inventor of input-output analysis is:
(a) Sir Francis Galton
(b) Fisher
(c) Prof. Wassily W. Leontief
(d) Arthur Cayley
(c) Prof. Wassily W. Leontief

Question 12.
Which of the following matrix has no inverse?
(a) $$\left(\begin{array}{rr} -1 & 1 \\ 1 & -4 \end{array}\right)$$
(b) $$\left(\begin{array}{rr} 2 & -1 \\ -4 & 2 \end{array}\right)$$
(c) $$\left(\begin{array}{cc} \cos a & \sin a \\ -\sin a & \cos a \end{array}\right)$$
(d) $$\left(\begin{array}{rr} \sin a & \sin a \\ -\cos a & \cos a \end{array}\right)$$
(b) $$\left(\begin{array}{rr} 2 & -1 \\ -4 & 2 \end{array}\right)$$
Hint:
So $$\left(\begin{array}{rr} 2 & -1 \\ -4 & 2 \end{array}\right)$$ has no inverse.

Question 13.
Inverse of $$\left(\begin{array}{ll} 3 & 1 \\ 5 & 2 \end{array}\right)$$ is:
(a) $$\left(\begin{array}{rr} 2 & -1 \\ -5 & 3 \end{array}\right)$$
(b) $$\left(\begin{array}{rr} -2 & 5 \\ 1 & -3 \end{array}\right)$$
(c) $$\left(\begin{array}{rr} 3 & -1 \\ -5 & -3 \end{array}\right)$$
(d) $$\left(\begin{array}{rr} -3 & 5 \\ 1 & -2 \end{array}\right)$$
(a) $$\left(\begin{array}{rr} 2 & -1 \\ -5 & 3 \end{array}\right)$$
Hint:
Let A = $$\left(\begin{array}{ll} 3 & 1 \\ 5 & 2 \end{array}\right)$$
|A| = [6 – 5] = 1
adj A = $$\left[\begin{array}{rr} 2 & -1 \\ -5 & 3 \end{array}\right]$$
∴ A-1 = $$\left[\begin{array}{rr} 2 & -1 \\ -5 & 3 \end{array}\right]$$

Question 14.
If A = $$\left(\begin{array}{rr} -1 & 2 \\ 1 & -4 \end{array}\right)$$ then A (adj A) is:
(a) $$\left(\begin{array}{ll} -4 & -2 \\ -1 & -1 \end{array}\right)$$
(b) $$\left(\begin{array}{rr} 4 & -2 \\ -1 & 1 \end{array}\right)$$
(c) $$\left(\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right)$$
(d) $$\left(\begin{array}{ll} 0 & 2 \\ 2 & 0 \end{array}\right)$$
(c) $$\left(\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right)$$
Hint:
A = $$\left(\begin{array}{rr} -1 & 2 \\ 1 & -4 \end{array}\right)$$
|A| = 4 – 2 = 2
We know that A (adj A) = |A| I
⇒ 2 $$\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)=\left(\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right)$$

Question 15.
If A and B non-singular matrix then, which of the following is incorrect?
(a) A2 = I implies A-1 = A
(b) I-1 = I
(c) If AX = B then X = B-1A
(d) If A is square matrix of order 3 then |adj A| = |A|2
(c) If AX = B then X = B-1A
Hint:
If AX = B then X = A-1B so, X = B-1A is incorrect.

Question 16.
The value of $$\left|\begin{array}{rrr} 5 & 5 & 5 \\ 4 x & 4 y & 4 z \\ -3 x & -3 y & -3 z \end{array}\right|$$ is:
(a) 5
(b) 4
(c) 0
(d) -3
(c) 0
Hint:
= 4 × (-3) $$\left|\begin{array}{lll} 5 & 5 & 5 \\ x & y & z \\ x & y & z \end{array}\right|$$
[Take out 4 from R2 and -3 from R3]
= 0 (∵ R2 ≡ R3)

Question 17.
If A is an invertible matrix of order 2 then det (A-1) be equal
(a) det (A)
(b) $$\frac{1}{{det}(A)}$$
(c) 1
(d) 0
(b) $$\frac{1}{{det}(A)}$$
Hint:
AA-1 = I
|AA-1| = |I|
|A| |A-1| = 1
|A-1| = $$\frac{1}{|\mathrm{A}|}$$
det A-1 = $$\frac{1}{\det (A)}$$

Question 18.
If A is 3 × 3 matrix and |A| = 4 then |A-1| is equal to:
(a) $$\frac{1}{4}$$
(b) $$\frac{1}{16}$$
(c) 2
(d) 4
(a) $$\frac{1}{4}$$
Hint:
|A-1| = $$\frac{1}{|A|}=\frac{1}{4}$$

Question 19.
If A is a square matrix of order 3 and |A| = 3 then |adj A| is equal to:
(a) 81
(b) 27
(c) 3
(d) 9
(d) 9
Hint:
|adj A| = |A|2 = 32 = 9

Question 20.
The value of $$\left|\begin{array}{ccc} x & x^{2}-y z & 1 \\ y & y^{2}-z x & 1 \\ z & z^{2}-x y & 1 \end{array}\right|$$ is:
(a) 1
(b) 0
(c) -1
(d) -xyz
(b) 0
Hint:

Question 21.
If A = $$\left[\begin{array}{rr} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$$, then |2A| is equal to:
(a) 4 cos 2θ
(b) 4
(c) 2
(d) 1
(b) 4
Hint:
|2A| = 22 |A|
= 4 $$\left|\begin{array}{rr} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right|$$
= 4 [cos2θ + sin2θ]
= 4 × 1
= 4

Question 22.
If Δ = $$\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|$$ and Aij is cofactor of aij, then value of Δ is given by:
(a) a11A31 + a12A32 + a13A33
(b) a11A11 + a12A21 + a13A31
(c) a21A11 + a22A12 + a23A13
(d) a11A11 + a21A21 + a31A31
(d) a11A11 + a21A21 + a31A31

Question 23.
If $$\left|\begin{array}{ll} x & 2 \\ 8 & 5 \end{array}\right|=0$$ then the value of x is:
(a) $$\frac{-5}{6}$$
(b) $$\frac{5}{6}$$
(c) $$\frac{-16}{5}$$
(d) $$\frac{16}{5}$$
(d) $$\frac{16}{5}$$
Hint:
$$\left|\begin{array}{ll} x & 2 \\ 8 & 5 \end{array}\right|=0$$
5x – 16 = 0
⇒ x = $$\frac{16}{5}$$

Question 24.
If $$\left|\begin{array}{ll} 4 & 3 \\ 3 & 1 \end{array}\right|$$ = -5 then the value of $$\left|\begin{array}{rr} 20 & 15 \\ 15 & 5 \end{array}\right|$$ is:
(a) -5
(b) -125
(c) -25
(4) 0
(b) -125
Hint:
$$\left|\begin{array}{rr} 20 & 15 \\ 15 & 5 \end{array}\right|$$
= 5 × 5 $$\left|\begin{array}{ll} 4 & 3 \\ 3 & 1 \end{array}\right|$$
= 5 × 5 × (-5)
= -125

Question 25.
If any three rows or columns of a determinant are identical then the value of the determinant is:
(a) 0
(b) 2
(c) 1
(d) 3