KC Sinha Mathematics Solution Class 12 Chapter 5 आव्यूह ( Matrices ) Exercise 5.2 (Q36-Q42)

Exercise 5.2


Question 36
(i) माना कि (Let) $f(x)=x^{2}-5 x+6$ (find) $f(A)$ निकाले यदि (if)
$A=\left[\begin{array}{lll}2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0\end{array}\right]$
Sol :
$f(a)=A^{2}-5 A+6 I$

(ii) यदि (If) $A=\left[\begin{array}{rrr}1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1\end{array}\right]$
दिखाएँ कि (show that) $A^{3}-23 A-40 I=0$
Sol :

(iii) यदि (If) $A=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]$, साबित करे कि  (prove that) $A^{3}-6 A^{2}+7 A+2 I=0$


(iv) यदि (If) $A=\left[\begin{array}{rr}3 & 1 \\ -1 & 2\end{array}\right]$, दिखाएँ कि (show that)
$A^{2}-5 A+7 I=O$
Sol :



Question 37
यदि (if) $A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ , साबित करे कि (prove that)
$A^{n}=\left[\begin{array}{ll}1 & n \\ 0 & 1\end{array}\right]$ सभी $n \in N$ के लिए (for all $n \in \mathbf{N}$)
Sol :
माना $P(n): A^{n}=\left[\begin{array}{ll}1 & n \\ 0 & 1\end{array}\right]$

$\begin{array}{rl}n=1 & A^{1}=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\end{array}$..(i)

P(1)=सत्य है ।

माना P(k) सत्य है

$P(k): A^{k}=\left[\begin{array}{ll}1 & k \\ 0 & 1\end{array}\right]$

तो साबित करना है कि P(k+1) भी सत्य होगा

$P(k+1): \quad A^{k+1}=\left[\begin{array}{cc}1 & k+1 \\ 0 & 1\end{array}\right]$

(i) मे दोनो तरफ A से गुना करने पर

$A^{k} \cdot A^{1}=\left[\begin{array}{ll}1 & k \\ 0 & 1\end{array}\right] \cdot A$

$A^{k+1}=\left[\begin{array}{ll}1 & k \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$

$=\left[\begin{array}{cc}1+0 & 1+k \\ 0+0 & 0-1\end{array}\right]$

$=\left[\begin{array}{ll}1 & k+1 \\ 0 & 1\end{array}\right]$

$n \in N$ के लिए $P(n): A^{n}=\left[\begin{array}{ll}1 & n \\ 0 & 1\end{array}\right]$ सत्य है ।


Question 38
यदि (if)$A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ दिखाएँ कि (show that) $A^{n}=\left[\begin{array}{cc}1+2 n & -4 n \\ n & 1-2 n\end{array}\right]$
जहाँ n एक धन पूर्णाक है (where n is a positive integer)
Sol :
माना P(n): $A^{n}=\left[\begin{array}{cc}1+2 n & -4 n \\ n & 1-2 n\end{array}\right]$

n=1 , $A^{\prime}=\left[\begin{array}{ccc}1+2(1) & -4(1) \\ 1 & 1-2(1)\end{array}\right]$

$=\left[\begin{array}{rr}3 & -4 \\ 1 & -1\end{array}\right]$

P(1) सत्य हैं 

माना P(k)  सत्य हैं 

P(k) : $A^{k}=\left[\begin{array}{cc}1+2 k & -4 k \\ k & 1-2 k\end{array}\right]$..(i)

तो , साबित करना है कि P(k+1) भी सत्य होगा ।

$A^{k+1}=\left[\begin{array}{cc}1+2(k-n) & -4(k+1] \\ k+1 & 1-2(k+1)\end{array}\right]$

(i) मे दोनो तरफ A से गुना करने पर ,

$A^{k} \cdot A=\left[\begin{array}{cc}1+2 k & -4 k \\ k & 1-2 k\end{array}\right]\left[\begin{array}{cc}3 & -4 \\ 1 & -1\end{array}\right]$

$A^{k+1}=\left[\begin{array}{cc}3+6 k-4 k & -4-8 k+4 k \\ 3 k+1-2 k & -4 k-1+2 k\end{array}\right]$

$A^{k-1}=\left[\begin{array}{cc}3+2 k & -4-4 k \\ k+1 & -2 k-1\end{array}\right]$

$A^{k+1}=\left[\begin{array}{cc}1+2 k+2 & -4 k-4 \\ k+1 & 1-2 k-2\end{array}\right]$

$=\left[\begin{array}{cc}1+2(k+1) & -4(k+1) \\ k+1 & 1-2(k+1)\end{array}\right]$

$\therefore n \in z^{+}$ के लिए $A^{n}=\left[\begin{array}{cc}1+2 n & -4 n \\ n & 1-2 n\end{array}\right]$ सत्य है ।



Question 39
यदि (if) A=diag.[a b c] दिखाएँ कि (show that) $\mathbf{A}^{a}=\operatorname{diag}\left[a^{n} \quad b^{n} \quad c^{n}\right]$ सभी (for all) $n \in \mathbf{N}$ के लिए
Sol :
माना $A^{n}=\left[\begin{array}{ccc}a^{n} & 0 & 0 \\ 0 & b^{n} & 0 \\ 0 & 0 & c^{n}\end{array}\right]$

n=1 , $A^{\prime}=\left[\begin{array}{lll}a^{1} & 0 & 0 \\ 0 & b^{1} & 0 \\ 0 & 0 & c^{1}\end{array}\right]$

$=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right]$

P(1) सत्य हैं 

माना P(k)  सत्य हैं 

P(k) : $A^{k}=\left[\begin{array}{lll}a^{k} & 0 & 0 \\ 0 & b^{k} & 0 \\ 0 & 0 & c^{k}\end{array}\right]$

तो , साबित करना है कि P(k+1) भी सत्य होगा ।

P(k+1): $A^{k+1}=\left[\begin{array}{ccc}a^{k+1} & 0 & 0 \\ 0 & b^{k+1} & 0 \\ 0 & 0 & c^{k+1}\end{array}\right]$

(i) मे दोनो तरफ A से गुना करने पर ,

$A^{k} \cdot A=\left[\begin{array}{lll}a^{k} & 0 & 0 \\ 0 & b^{k} & 0 \\ 0 & 0 & c^{k}\end{array}\right]\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right]$

$A^{k+1}=\left[\begin{array}{ccc}a^{k+1} & 0 & 0 \\ 0 & b^{k+1} & 0 \\ 0 & 0 & c^{k-1}\end{array}\right]$

$\therefore n \in z^{+}$ के लिए  $A^{n}=\left[\begin{array}{lll}a^{n} & 0 & 0 \\ 0 & b^{n} & 0 \\ 0 & a & c^{1}\end{array}\right]$ सत्य है ।


Question 40
यदि (if) $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$ ,
तो गणितीय आगमन सिद्धान्त से साबित करे कि (prove by principle of mathematical induction that)
$A^{n}=\left[\begin{array}{cc}\cos n \alpha & \sin n \alpha \\ -\sin n \alpha & \cos n \alpha\end{array}\right]$
प्रत्येक प्राकृत संख्या n के लिए (for every natural number n)
Sol :
Let P(n): $A^{n}=\left[\begin{array}{cc}\cos n \alpha & \sin n \alpha \\ -\sin n \alpha & \cos n \alpha\end{array}\right]$

when n=1 ,

$A^{1}=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$

P(1) is true

Let P(k) be true 

$P(k): A^{k}=\left[\begin{array}{cc}\cos k \alpha & \sin k \alpha \\ -\sin k \alpha & \cos k \alpha\end{array}\right]$

then prove that P(k+1) is true

P(k+1):
$A^{k+1}=\left[\begin{array}{cc}\cos (k+1) \alpha & \sin (k+1) \alpha \\ -\sin (k+1) \alpha & \cos (k+1) \alpha\end{array}\right]$

<to be added>

$A^{k} \cdot A=\left[\begin{array}{cc}\cos k \alpha & \sin k \alpha \\ -\sin k \alpha & \cos k \alpha\end{array}\right]\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$

$A^{k+1}=\left[\begin{array}{ll}\cos k \alpha+\cos \alpha-\sin \alpha \sin \alpha & cosk\alpha \sin \alpha+ \sin k \alpha \cos \alpha\\ -\cos \alpha \sin k \alpha-\cos k\alpha\sin \alpha & -\sin k \alpha\sin \alpha+\cos k\alpha \cos \alpha\end{array}\right]$

$=\left[\begin{array}{ll}\cos (k\alpha+\alpha) & \sin (k \alpha+\alpha) \\ -\sin (k\alpha+\alpha) & \cos (k+\alpha)\end{array}\right]$

$A^{k+1}=\left[\begin{array}{cc}\cos (k+1) \alpha & \sin (k+1)+\alpha \\ -\sin (k-1) \alpha & \cos (k+1) \alpha \end{array}\right]$

∴ nϵN के लिए , $A^{n}=\left[\begin{array}{cc}\cos n \alpha & \sin \alpha \\ -\sin n \alpha & \cos n \alpha\end{array}\right]$ सत्य है ।


Question 41

यदि (If) $A=\left[\begin{array}{ll}\cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta\end{array}\right]$ , तो गणितीय आगमन सिद्धान्त से साबित करें कि (then prove by principle f mathematical induction that) $A^{n}=\left[\begin{array}{cc}\cos n \theta & i \sin n \theta \\ i \sin n \theta & \cos n \theta\end{array}\right]$ , जहाँ (where) n∊N
Sol :




Question 42

यदि A और B समान कोटिवाले वर्ग आव्यूह इस प्रकार है कि AB=BA , तो गणितीय आगमन सिद्धान्त से साबित करे कि $AB^n=B^nA$ . इसके अतिरिक्त यह भी साबित करें कि $(\mathrm{AB})^{n}=\mathrm{A}^{n} \mathrm{~B}^{n}$ सभी n∈N के लिए।

[If A and B are square matrices of the smae order such that AB=BA then prove by principle of mathematical induction that $AB^n=B^nA$. Further, prove that $(\mathrm{AB})^{n}=\mathrm{A}^{n} \mathrm{~B}^{n}$ for all n∈N ]
Sol :

Let P(n): $A B^{n}=B^{n} A$
where n∈N

when n=1 ,

$A B^{\prime}=B^{\prime} A$⇒AB=BA

P(1) is true

Let P(k) be true

$P(k): A B^{k}=B^{k} A$..(i)

then prove that P(k+1) is also true

$A B^{k+1}=B^{k+1} \cdot A$
Multiplying both side by B in eq-(i)

$A B^{k} \cdot B^{1}=B^{k} A \cdot B^{1}$

$A B^{\prime \prime+1}=B^{k} A B$

$A B^{k+1}=B^{k} \cdot B^{1} A$

$A B^{k+1}=B^{k+1} \cdot A$

On n=1 ,P(n) be true


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