KC Sinha Mathematics Solution Class 12 Chapter 5 आव्यूह ( Matrices ) Exercise 5.1 (Q21-Q30)

Ex-5.1

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Exercise 5.1

Question 21
यदि (if) $A=\left[\begin{array}{rrr}1 & 2 & 3 \\ -1 & 0 & 2 \\ 1 & -3 & -1\end{array}\right], B=\left[\begin{array}{rrr}4 & 5 & 6 \\ -1 & 0 & 1 \\ 2 & 1 & 2\end{array}\right]$ तथा (and) $C=\left[\begin{array}{rrr}-1 & -2 & 1 \\ -1 & 2 & 3 \\ -1 & -2 & 2\end{array}\right]$

A+(B+C)=(A+B)+C को सत्यापित करे ।
[verify that A+(B+C)=(A+B)+C]
Sol :
L.H.S
A+(B+C)

$=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 0 & 2 \\ 1 & -3 & -1\end{array}\right]+\left(\left[\begin{array}{ccc}4 & 5 & 6 \\ -1 & 0 & 1 \\ 2 & 1 & 2\end{array}\right]+\left[\begin{array}{ccc}-1 & -2 & 1 \\ -1 & 2 & 3 \\ -1 & -2 & 2\end{array}\right]\right)$

$=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 0 & 2 \\ 1 & -3 & -1\end{array}\right]+\left[\begin{array}{ccc}3 & 3 & 7 \\ -2 & 2 & 4 \\ 1 & -1 & 4\end{array}\right]$

$=\left[\begin{array}{ccc}4 & 5 & 10 \\ -3 & 2 & 6 \\ 2 & -4 & 3\end{array}\right]$

R.H.S
(A+B)+C

$=\left(\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 0 & 2 \\ 1 & -3 & -1\end{array}\right]+\left[\begin{array}{ccc}4 & 5 & 6 \\ -1 & 0 & 1 \\ 2 & 1 & 2\end{array}\right]\right)+\left[\begin{array}{ccc}-1 & -2 & 1 \\ -1 & 2 & 3 \\ -1 & -2 & 2\end{array}\right]$

$=\left[\begin{array}{ccc}5 & 7 & 9 \\ -2 & 0 & 3 \\ 3 & -2 & 1\end{array}\right]+\left[\begin{array}{ccc}-1 & -2 & 1 \\ -1 & 2 & 3 \\ -1 & -2 & 2\end{array}\right]$

$=\left[\begin{array}{ccc}4 & 5 & 10 \\ -3 & 2 & 6 \\ 2 & -4 & 3\end{array}\right]$

L.H.S=R.H.S


Question 22
निकाले (Evaluate) $\begin{bmatrix}\sin ^{2} \theta & 1\\\cot ^{2} \theta & 0\end{bmatrix}+\left[\begin{array}{cc}\cos ^{2} \theta & 0 \\ -\operatorname{cosec}^{2} \theta & 1\end{array}\right]+\left[\begin{array}{cc}0 & -1 \\ -1 & 0\end{array}\right]$
Sol :
$=\left[\begin{array}{ll}\sin ^{2} \theta+cos^{2} \theta+0 & 1+0-1 \\ \cot ^{2} \theta-cosec ^{2} \theta-1 & 0+1+0\end{array}\right]$

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

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


Question 23
(i) निम्नलिखित समीकरण से x और y निकाले
[From the following equations , find the values of x and y]
$2\left[\begin{array}{cc}x & 5 \\ 7 & y-3\end{array}\right]+\left[\begin{array}{cc}3 & 4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 14 \\ 15 & 14\end{array}\right]$
Sol :
$\left[\begin{array}{cc}2 x & 10 \\ 14 & 2 y-6\end{array}\right]+\left[\begin{array}{cc}3 & 4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 14 \\ 15 & 14\end{array}\right]$

$\left[\begin{array}{cc}2 x+3 & 14 \\ 15 & 2 y-4\end{array}\right]=\left[\begin{array}{cc}7 & 14 \\ 15 & 14\end{array}\right]$

⇒2x+3=7
⇒2x=4
⇒x=2

⇒2y-4=14
⇒2y=18
⇒y=9

(ii) निम्नलिखित समीकरण को संतुष्ट करने वाले x और y का मान निकालें। 

[Find the values of x and y satisfying the following equation]

$2\left[\begin{array}{cc}x & 5 \\ 7 & y-3\end{array}\right]+\left[\begin{array}{lr}3 & -4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 6 \\ 15 & 14\end{array}\right]$

Sol :

Question 24

निम्नलिखित समीकरण को संतुष्ट करने वले x, y, z तथा t का मान निकालें । 

[Find x, y, z and t satisfying the following.equation] :

$2\left[\begin{array}{ll}x & z \\ y & t\end{array}\right]+3\left[\begin{array}{lr}1 & -1 \\ 0 & 2\end{array}\right]=3\left[\begin{array}{ll}3 & 5 \\ 4 & 6\end{array}\right]$

Sol :

Question 25
यदि A=विकर्ण [1 2  3]
B=विकर्ण [0 2  5]
C=विकर्ण [3 -2  5]

If A=diag [1 2  3]
B=diag [0 2  5]
C=diag [3 -2  5]
 

(i)4A-3B
Sol :
=diag[4  12  8]-diag[0  6  15]
=diag[4  6  -7]


(ii) A+B-2C
Sol :


Question 26
आव्यूह X निकालें यदि (Find matrix X , if ) $X+\left[\begin{array}{ll}2 & 5 \\ 3 & 2\end{array}\right]=\left[\begin{array}{rr}4 & 0 \\ -7 & 6\end{array}\right]$
Sol :
$x=\left[\begin{array}{cc}4 & 0 \\ -7 & 6\end{array}\right]-\left[\begin{array}{ll}2 & 5 \\ 3 & 2\end{array}\right]$

$x=\left[\begin{array}{rr}2 & -5 \\ -10 & 4\end{array}\right]$


Question 27
आव्यूह X निकालें ताकि (Find a matrix X such that) 2A+B+X=0
जहाँ (where) $A=\left[\begin{array}{rr}-1 & 2 \\ 3 & 4\end{array}\right]$  तथा (and) $B=\left[\begin{array}{rr}3 & -2 \\ 1 & 5\end{array}\right]$
Sol :
2A+B+X=0

$2\left[\begin{array}{cc}-1 & 2 \\ 3 & 4\end{array}\right]+\left[\begin{array}{cc}3 & -2 \\ 1 & 5\end{array}\right]+X=0$

$\left[\begin{array}{cc}-2 & 4 \\ 6 & 8\end{array}\right]+\left[\begin{array}{cc}3 & -2 \\ 1 & 5\end{array}\right]+X=0$

$\left[\begin{array}{ll}1 & 2 \\ 7 & 13\end{array}\right]+X=0$

$X=\left[\begin{array}{cc}-1 & -2 \\ -7 & -13\end{array}\right]$


Question 28
(i) आव्यूह X निकालें ताकि ( Find a matrix X such that )

A+2B+X=0 ( जहाँ ) where
$A=\left[\begin{array}{rr}2 & -1 \\ 3 & 5\end{array}\right] ; B=\left[\begin{array}{rr}-1 & 1 \\ 0 & 2\end{array}\right]$

(ii)  3×2 कोटि का एक आव्यूह X निकालें ताकि ( Find a matrix X of order 3×2 such that) 2A+3X=5 , जहाँ (where) $A=\left[\begin{array}{rr}8 & 0 \\ 4 & -2 \\ 3 & 6\end{array}\right]$ तथा (and) $B=\left[\begin{array}{rr}2 & -2 \\ 4 & 2 \\ -5 & 1\end{array}\right]$

Sol :

Question 29
आव्यूह X तथा Y निकालें यदि (Find matrices X and Y , if) $x+y=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]$ तथा (and) $\mathrm{X}-\mathrm{Y}=\left[\begin{array}{ll}3 & \phantom{-}6 \\ 0 & -1\end{array}\right]$
Sol :
$\begin{aligned} X+Y &=\left[\begin{array}{ll}5 & 2 \\ 0 & 3\end{array}\right]..(i) \\ X-Y &=\left[\begin{array}{ll}3 & \phantom{-}6 \\ 0 & -1\end{array}\right] ..(ii)\\\hline 2 x &=\left[\begin{array}{ll}8 & 8 \\ 0 & 8\end{array}\right] \end{aligned}$

$X=\left[\begin{array}{cc}4 & 4 \\ 0 & 4\end{array}\right]$

Putting the value of X in equation (i)

$\left[\begin{array}{ll}4 & 4 \\ 0 & 4\end{array}\right]+Y=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]$

$y=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]-\left[\begin{array}{cc}4 & 4 \\ 0 & 4\end{array}\right]$

$y=\left[\begin{array}{cc}1 & -2 \\ 0 & 5\end{array}\right]$


Question 30
दिखाएँ कि (Show that)
$\cos \theta\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]+\sin \theta\left[\begin{array}{cr}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta\end{array}\right]=I$
Sol :
L.H.S
$\cos \theta\left[\begin{array}{rr}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]+\sin \theta\left[\begin{array}{rr}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta\end{array}\right]$

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

$=\left[\begin{array}{ccc}\cos ^{2} \theta+sin^2 \theta & 0 \\ 0 & \cos ^{2} \theta+\sin ^{2} \theta\end{array}\right]$

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