KC Sinha Mathematics Solution Class 12 Chapter 5 आव्यूह ( Matrices ) Exercise 5.3

Ex-5.3

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Exercise 5.3

Question 1

आव्यूह $\left[\begin{array}{rr}1 & 3 \\ 2 & 6 \\ 5 & -3\end{array}\right]$ का परिवर्त ज्ञात करें।

[Find the transpose of the matrix $\left[\begin{array}{rr}1 & 3 \\ 2 & 6 \\ 5 & -3\end{array}\right]$ ]

Sol :

Question 2

यदि (if) $A=\left[\begin{array}{r}3 \\ 1 \\ -2\end{array}\right]$ , $B=\left[\begin{array}{lll}1 & -5 & 7\end{array}\right]$ तो सत्यापित करें (verify that) (AB)'=B'A'
Sol :
$A B=\left[\begin{array}{c}3 \\ -1 \\ -2\end{array}\right]\left[\begin{array}{lll}1 & -5 & 7\end{array}\right]$

$=\left[\begin{array}{ccc}3 & -15 & 21 \\ 1 & -5 & 7 \\ -2 & 10 & -14\end{array}\right]$

$\Rightarrow(A B)'=\left[\begin{array}{ccc}3 & 1 & -2 \\ -15 & -5 & 10 \\ 21 & 7 & -14\end{array}\right]$

$B^{\prime} A^{\prime}=\left[\begin{array}{c}1 \\ -5 \\ 7\end{array}\right]\left[\begin{array}{lll}3 & 1 & -2\end{array}\right]$

(AB)'=B'A'

Question 3

(i) यदि (If) $\mathrm{A}=\left[\begin{array}{c}-2 \\ 4 \\ 5\end{array}\right], \mathrm{B}=\left[\begin{array}{lll}1 & 3 & -5\end{array}\right]$ तो सत्यापित करें कि (verify that) (AB)'=B'A'

(ii) यदि (If) $\mathrm{A}=\left[\begin{array}{r}-1 \\ 2 \\ 3\end{array}\right]$ तथा (and) $\mathrm{B}=\left[\begin{array}{lll}-2 & -1 & -4\end{array}\right]$, तो सत्यापित करें कि (verify that) (AB)'=B'A'

(iii) यदि (If) $A=\left[\begin{array}{l}3 \\ 5 \\ 2\end{array}\right]$ तथा (and) $B=\left[\begin{array}{lll}1 & 0 & 4]\end{array}\right]$ तो सत्यापित करें कि (verify that) (AB)'=B'A'

Question 4

(i) यदि (If) $\mathrm{A}=\left[\begin{array}{lll}2 & 1 & 3 \\ 4 & 1 & 0\end{array}\right]$ तथा (and) $\mathrm{B}=\left[\begin{array}{rr}1 & -1 \\ 0 & 2 \\ 5 & 0\end{array}\right]$, तो सत्यापित करें कि (verify that) (AB)'=B'A'

(ii) यदि [If] $A=\left[\begin{array}{ll}1 & 4 \\ 0 & 5 \\ 6 & 7\end{array}\right]$ तथा (and) $B=\left[\begin{array}{rrr}2 & 3 & -1 \\ 1 & 0 & -7\end{array}\right]$, तो सत्यापित करें कि (verify that) (AB)'=B'A'

(iii) यदि (If) $\mathrm{A}=\left[\begin{array}{lll}2 & 1 & 3 \\ 4 & 1 & 0\end{array}\right], \mathrm{B}=\left[\begin{array}{lr}1 & -1 \\ 0 & 2 \\ 5 & 0\end{array}\right]$, तो सत्यापित करें कि (verify that) (AB)'=B'A'

(iv) यदि (If) $\mathrm{A}=\left[\begin{array}{rr}3 & -4 \\ 1 & 1 \\ 1 & 0\end{array}\right]$ तथा (and) $\mathrm{B}=\left[\begin{array}{lll}2 & 1 & 2 \\ 1 & 3 & 4\end{array}\right]$, तो सत्यापित करें कि (verify that) (AB)'=B'A'

Question 5
यदि (if) $A=\left[\begin{array}{rr}5 & -1 \\ 6 & 7\end{array}\right]$ ,$B=\left[\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right]$ ,$C=\left[\begin{array}{rr}1 & 3 \\ -1 & 4\end{array}\right]$ तो निम्नलिखित को सत्यापित कीजिए , (verify the following)
(i) (A+B)'=A'+B'
(ii) (3B)'=3B'
(iii) (A')'=A
(iv) (AV)'=C'A'
(v) (AB)'=B'A'
Sol :
L.H.S
(i)
L.H.S
(A+B)'=$\left[\begin{array}{ll}7 & 0 \\ 10 & 11\end{array}\right]^{\prime}$

$=\left[\begin{array}{ll}7 & 10 \\ 0 & 11\end{array}\right]$

R.H.S
$A^{\prime}+B^{\prime}=\left[\begin{array}{cc}5 & 6 \\ -1 & 7\end{array}\right]+\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right]$

$=\left[\begin{array}{ll}7 & 10 \\ 0 & 1\end{array}\right]$

(A+B)'=A'+B'


Question 6
(i) यदि (if)  $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$  , तो दिखाएँ कि (show that)
A'A=I₂
Sol :
L.H.S
$A^{\prime} A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$

$=\left[\begin{array}{ll}\cos ^{2} \alpha+\sin ^{2} \alpha & \cos \alpha \sin \alpha-\sin \alpha \cos \alpha \\ \sin \alpha \cos \alpha-\sin \alpha \cos \alpha & \sin ^{2} \alpha+\cos ^{2} \alpha \end{array}\right]$

$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=I_{2}$

(ii) यदि (If) $A=\left[\begin{array}{rr}\sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha\end{array}\right]$, तो सत्यापित करें कि (verify that) A'A=I

Sol :

Question 7
x और y ज्ञात करे यदि आव्यूह [find x and y if the matrix]
$A=\frac{1}{3}\left[\begin{array}{llr}1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y\end{array}\right]$ शर्त (satisfy the condition) AA'=A'A=I₃ को संतुष्ट करता है ।
Sol :
AA'=I₃

$\frac{1}{3}\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y\end{array}\right] \cdot \frac{1}{3}\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & -2 & y\end{array}\right]=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

$\frac{1}{9}\left[\begin{array}{ccc}1+4+4 & 2+2-4 & x+4+2 y \\ 2+2-4 & 4+1+4 & 2 x+2-2 y \\ x+4+2 y & 2 x+2-2 y & x^{2}+4+y^{2}\end{array}\right]=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

$\left[\begin{array}{ccc}9 & 0 & x+2 y+4 \\ 0 & 9 & 2 x-2 y+2 \\ x+2 y+4 & 2 x-2 y+2 & x^{2}+y^{2}+4\end{array}\right]=\left[\begin{array}{ccc}9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9\end{array}\right]$

x+2y+4=0 , 2x-2y+2=0

x+2y=-4..(i) , 2x-2y=-2..(ii)

समीo (i) तथा (ii) ,

$\begin{aligned} x+2 y &=-4 \\ 2 x-2 y &=-2 \\ \hline 3 x &=-6 \end{aligned}$

x का मान समीo (i) मे x=-2

⇒x+2y=-4
⇒-2+2y=-4
⇒2y=-2
⇒y=-1

∴x=-2 , y=-1