NCERT Solutions Class 12- Mathematics Part I - Chapter 4 Determinants - Miscellaneous Exercises on Chapter 4

Question 1. Prove that the determinant \begin{vmatrix} x & sin\theta  & cos\theta \\ -sin\theta  & -x & 1\\ cos\theta  & 1 & x \end{vmatrix}      is independent of θ.

Solution: 

A = \begin{vmatrix} x & sin\theta  & cos\theta \\ -sin\theta  & -x & 1\\ cos\theta  & 1 & x \end{vmatrix}

A = x(x² - 1) - sinθ(-x sinθ - cosθ) + cosθ(-sinθ + x cosθ)

A = x³ - x + x sin2θ + sinθcosθ - sinθcosθ + x cos2θ

A = x³ - x + x(sin2θ + cos2θ)

A = x³ - x + x

A = x³(Independent of θ).

Hence, it is independent of θ

Question 2. Evaluate \begin{vmatrix} cos\alpha cos\beta   & cos\alpha sin\beta  & -sin\alpha\\ -sin\beta & cos\beta & 0\\ sin\alpha cos\beta & sin\alpha sin\beta & cos\alpha \end{vmatrix}

Solution: 

A = \begin{vmatrix} cos\alpha cos\beta   & cos\alpha sin\beta  & -sin\alpha\\ -sin\beta & cos\beta & 0\\ sin\alpha cos\beta & sin\alpha sin\beta & cos\alpha \end{vmatrix}

Expanding along C3 

A = -sinα(-sinα sin²β - cos²β sinα) + cosα(cosα cos²β + cosα sin²β)

A = sin²α(sin²β + cos²β) + cos²α(cos²β + sin²β)

A = sin²(1) + cos²(1)

A = 1

Question 3. If A^-1 =\begin{vmatrix} 3 & -1 & 1\\ -15 & 6 &-5\\ 5 & -2 & 2 \end{vmatrix}     and B =\begin{vmatrix} 1 & 2 & -2\\ -1 & 3 &0\\ 0 & -2 & 1 \end{vmatrix}     . Find (AB)^-1

Solution:

|B| = 1(3 - 0) + 1(2 - 4) = 1

B₁₁ = 3 - 0 = 3

B₁₂ = 1

B₁₃ = 2 - 0 = 2

B₂₁ = -(2 - 4) = 2

B₂₂ = 1 - 0 = 1

B₂₃ = 2

B₃₁ = 0 + 6 = 6

B₃₂ = -(0 - 2) = 2

B₃₃ = 3 + 2 = 5

adj B = \begin{bmatrix} 3 & 2 & 6\\ 1 & 1 & 2\\ 2 & 2 & 5 \end{bmatrix}

B^-1 = (adj B)/|B|

B^-1 = \begin{bmatrix} 3 & 2 & 6\\ 1 & 1 & 2\\ 2 & 2 & 5 \end{bmatrix}

Now,

(AB)^-1 = B^-1A^-1

(AB)^-1 = \begin{bmatrix} 3 & 2 & 6\\ 1 & 1 & 2\\ 1 & 2 &5 \end{bmatrix} \begin{bmatrix} 3 & -1 & 1\\ -15 & 6 & -5\\ 5 & -2 & 2 \end{bmatrix}

= \begin{bmatrix} 9-30+30 & -3+12-12 & 3-10+12\\ 3-15+10 & -1+6-4 & 1-5+4\\ 6+12-10 & -2+12-10 & 2-10+10 \end{bmatrix}

(AB)^-1 = \begin{bmatrix} 9 & -3 & 5\\ -2 & 1 & 0\\ 8 & 0 & 2 \end{bmatrix}

Question 4. Let A =\begin{bmatrix} 1 & -2 & 1\\ -2 & 3 & 1\\ 1 & 1 & 5 \end{bmatrix}     verify that

(i) [adj A]^-1 = adj(A^-1)

(ii) (A^-1)^-1 = A

Solution: 

A = \begin{bmatrix} 1 & -2 & 1\\ -2 & 3 & 1\\ 1 & 1 & 5 \end{bmatrix}

|A| = 1(15 - 1) + 2(-10 - 1) + 1(-2 - 3) = 14 - 27 = -13

A₁₁ = 14

A₁₂ = 11

A₁₃ = -5

A₂₁ = 11

A₂₂ = 4

A₂₃ = -3

A₃₁ = -5

A₃₂ = -3

A₃₃ = -1

adj A = \begin{bmatrix} 14 & 11 & -5\\ 11 & 4 & -3\\ -5 & -3 & -1 \end{bmatrix}

Arrr^-1 = (adj A)/|A|

= -1/13\begin{bmatrix} 14 & 11 & -5\\ 11 & 4 & -3\\ -5 & -3 & -1 \end{bmatrix}

= 1/13\begin{bmatrix} -14 & -11 & 5\\ -11 & -4 & 3\\ 5 & 3 & 1 \end{bmatrix}

(i). |adj A| = 14(-4 - 9) - 11(-11 - 15) - 5(-33 + 20)

= 14(-13) - 11(-26) - 5(-13)

= -182 + 286 + 65 = 169

adj(adj A) = \begin{bmatrix} -13 & 26 & -13\\ 26 & -39 & -13\\ -13 & -13 & -65 \end{bmatrix}

[adj A]^-1 = (adj(adj A))/|adj A|

= 1/169\begin{bmatrix} -13 & 26 & -13\\ 26 & -39 & -13\\ -13 & -13 & -65 \end{bmatrix}

= 1/13\begin{bmatrix} -1 & 2 & -1\\ 2 & -3 & -1\\ -1 & -1 & -5 \end{bmatrix}

Now, A^-1 = 1/13\begin{bmatrix} -14 & -11 & 5\\ -11 & -4 & 3\\ 5 & 3 & 1 \end{bmatrix}

= \begin{bmatrix} \frac{-14}{13} & \frac{-11}{13} & \frac{5}{13}\\ \frac{-11}{13} & \frac{-4}{13} & \frac{3}{13}\\ \frac{5}{13} & \frac{3}{13} & \frac{1}{13} \end{bmatrix}

adj(A^-1) = \begin{bmatrix} \frac{-4}{169}-\frac{9}{169} & -(\frac{-11}{169}-\frac{15}{169}) & \frac{-33}{169}+\frac{20}{169}\\ -(\frac{-11}{169}-\frac{-15}{169}) & \frac{-14}{169}-\frac{25}{169} &-(\frac{-42}{169}+\frac{55}{169}) \\ \frac{-33}{169}+\frac{20}{169} & -(\frac{-42}{169}+\frac{55}{169}) & \frac{56}{169}-\frac{121}{169} \end{bmatrix}

= 1/169\begin{bmatrix} -13 & 26 & -13\\ 26 & -39 & -13\\ -13 & -13 & -65 \end{bmatrix}

= 1/13\begin{bmatrix} -1 & 2 & -1\\ 2 & -3 & -1\\ -1 & -1 & -5 \end{bmatrix}

Hence, [adj A]^-1 = adj(A^-1)

(ii). A^-1 = 1/13\begin{bmatrix} -14 & -11 & 5\\ -11 & -4 & 3\\ 5 & 3 & 1 \end{bmatrix}

adj A^-1 = 1/13\begin{bmatrix} -1 & 2 & -1\\ 2 & -3 & -1\\ -1 & -1 & -5 \end{bmatrix}

|A^-1| = (1/13)3[-14 × (-13) +11 × (-26) + 5 × (-13)]

= (1/13)3 × (-169)

= -1/13

Now, (A^-1)^-1 = (adj A^-1)/|A^-1|

= \frac{1}{(\frac{-1}{13})}*\frac{1}{13}\begin{bmatrix} -1 & 2 & -1\\ 2 & -3 & -1\\ -1 & -1 & -5 \end{bmatrix}

= \begin{bmatrix} 1 & -2 & 1\\ -2 & 3 & 1\\ 1 & 1 & 5 \end{bmatrix}

= A

Hence, it is proved that (A^-1)^-1 = A

Question 5. Evaluate \begin{vmatrix} x & y & x+y\\ y & x+y & x\\ x+y & x &y \end{vmatrix}

Solution: 

A = \begin{vmatrix} x & y & x+y\\ y & x+y & x\\ x+y & x &y \end{vmatrix}

Applying R1 -> R1+R2+R3

A = \begin{vmatrix} 2(x+y) & 2(x+y) & 2(x+y)\\ y & x+y & x\\ x+y & x &y \end{vmatrix}

= 2(x+y)\begin{vmatrix} 1 & 1 & 1\\ y & x+y & x\\ x+y & x &y \end{vmatrix}

Applying C2-> C2 - C1 and C3-> C3 - C1

A = 2(x + y)\begin{vmatrix} 1 & 0 & 0\\ y & x & x-y\\ x+y & -y &-x \end{vmatrix}

Expanding along R1

A = 2(x + y)[-x² + y(x - y)]

= -2(x + y)(x² + y² - yx)

A = -2(x³ + y³)

Question 6. Evaluate \begin{vmatrix} 1 & x & y\\ 1 & x+y & y\\ 1 & x &x+y \end{vmatrix}

Solution:

A = \begin{vmatrix} 1 & x & y\\ 1 & x+y & y\\ 1 & x &x+y \end{vmatrix}

Applying R2->R2 - R1 and R3->R3 - R1

A = \begin{vmatrix} 1 & x & y\\ 0 & y & 0\\ 0 & 0 &x \end{vmatrix}

Expanding along C1

A = 1(xy - 0)

A = xy

Question 7. Solve the system of the following questions:

2/x + 3/y + 10/z = 4

4/x - 6/y + 5/z = 1

6/x + 9/y - 20/z = 2

Solution:

Assume 1/x = p ; 1/y = q; 1/z = r

then. the above equations will be like

2p + 3Q + 10r = 4

4p - 6q + 5r = 1

6p + 9q - 20r = 2

This can be written in the form of AX=B

where,

A = \begin{bmatrix} 2 & 3 & 10\\ 4 & -6 & 5\\ 6 & 9 & -20 \end{bmatrix}

X = \begin{bmatrix} p\\ q\\ r \end{bmatrix}

B = \begin{bmatrix} 4\\ 1\\ 2 \end{bmatrix}

We have,

|A| = 2(120 - 45) - 3(-80 - 30) + 10(36 + 36)

|A| = 150 + 330 + 720

|A| = 1200 ≠ 0

Hence A is invertible matrix.

A₁₁ = 75

A₁₂ = 110

A₁₃ = 72

A₂₁ = 150

A₂₂ = -100

A₂₃ = 0

A₃₁ = 75

A₃₂ = 30

A₃₃ = -24

A^-1 = (adj A)/|A|

A^-1 = \frac{1}{1200}\begin{bmatrix} 75 & 150 & 75\\ 110 & -100 & 30\\ 72 & 0 & -24 \end{bmatrix}

Now,

X = A^-1B

\begin{bmatrix} p\\ q\\ r \end{bmatrix}   = \frac{1}{1200}\begin{bmatrix} 75 & 150 & 75\\ 110 & -100 & 30\\ 72 & 0 & -24 \end{bmatrix} \begin{bmatrix} 4\\ 1\\ 2 \end{bmatrix}

\begin{bmatrix} p\\ q\\ r \end{bmatrix}   = \frac{1}{1200}\begin{bmatrix} 300+150+150\\ 440-100+60\\ 288+0-48 \end{bmatrix}

= \frac{1}{1200}\begin{bmatrix} 600\\ 400\\ 240 \end{bmatrix}

= \begin{bmatrix} \frac{1}{2}\\ \frac{1}{3}\\ \frac{1}{5} \end{bmatrix}

From above p = 1/2; q = 1/3 ; r = 1/5

So, x = 2; y = 3; z = 5

Question 8. Choose the correct answer.

If x, y, z are non-zero real numbers, then the inverse of matrix A = \begin{bmatrix} x & 0 & 0\\ 0 & y &0 \\ 0 & 0 & z \end{bmatrix}    is 

(A) \begin{bmatrix} x^{-1} & 0 & 0\\ 0 & y^{-1} &0 \\ 0 & 0 & z^{-1} \end{bmatrix}

(B) xyz\begin{bmatrix} x^{-1} & 0 & 0\\ 0 & y^{-1} &0 \\ 0 & 0 & z^{-1} \end{bmatrix}

(C) \frac{1}{xyz}\begin{bmatrix} x & 0 & 0\\ 0 & y &0 \\ 0 & 0 & z \end{bmatrix}

(D) \frac{1}{xyz}\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 &0 \\ 0 & 0 & 1 \end{bmatrix}

Solution:

A = \begin{bmatrix} x & 0 & 0\\ 0 & y &0 \\ 0 & 0 & z \end{bmatrix}

|A| = x(yz - 0) = xyz ≠ 0

Hence, the matrix is invertible

Now,

A₁₁ = yz

A₁₂ = 0

A₁₃ = 0

A₂₁ = 0

A₂₂ = xz

A₂₃ = 0

A₃₁ = 0

A₃₂ = 0

A₃₃ = xy

adj A = \begin{bmatrix} yz & 0 & 0\\ 0 & xz &0 \\ 0 & 0 & xy \end{bmatrix}

A^-1 = (adj A)/|A|

A^-1 = \frac{1}{xyz}\begin{bmatrix} yz & 0 & 0\\ 0 & xz &0 \\ 0 & 0 & xy \end{bmatrix}

A^-1 = \begin{bmatrix} \frac{yz}{xyz} & 0 & 0\\ 0 & \frac{xz}{xyz} &0 \\ 0 & 0 & \frac{xy}{xyz} \end{bmatrix}

A^-1 = \begin{bmatrix} \frac{1}{x} & 0 & 0\\ 0 & \frac{1}{y} &0 \\ 0 & 0 & \frac{1}{z} \end{bmatrix}

A^-1 = \begin{bmatrix} x^{-1} & 0 & 0\\ 0 & y^{-1} &0 \\ 0 & 0 & z^{-1} \end{bmatrix}

Hence, the correct answer is A.

Question 9. Choose the correct answer

Let A = \begin{bmatrix} 1 & sin\theta  & 1\\ -sin\theta & 1 & sin\theta\\ -1 & -sin\theta & 1 \end{bmatrix}   , where 0 ≤ θ ≤ 2π, then

(A) Det(A) = 0                                      (B) Det(A) ∈ (2, ∞)

(C) Det(A) ∈ (2, 4)                              (D) Det(A) ∈ [2, 4]

Solution:

A = \begin{bmatrix} 1 & sin\theta  & 1\\ -sin\theta & 1 & sin\theta\\ -1 & -sin\theta & 1 \end{bmatrix}

|A| = 1(1 + sin2θ) - sinθ(-sinθ + sinθ) + 1(sin2θ + 1)

|A| = 1 + sin2θ + sin2θ + 1

= 2 + 2 sin2θ

= 2(1 + sin2θ)

Now 0 ≤ θ ≤ 2π

So, 0 ≤ sinθ ≤ 1

0 ≤ sin2θ ≤ 1

0 + 1 ≤ 1 + sin2θ ≤ 1 + 1

2 ≤ 2(1 + sin2θ) ≤ 4

Det(A) ∈ [2, 4]

Hence, the correct answer is D.