KC Sinha Mathematics Solution Class 12 Chapter 5 आव्यूह ( Matrices ) Exercise 5.2 (Q29-Q35)
Exercise 5.2
Question 29
आव्यूह X निकालें ताकि
(Find the matrix X such that)
$X\left[\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right]=\left[\begin{array}{rr}0 & -4 \\ 10 & 3\end{array}\right]$
Sol :
Question 30
यदि (if) $A=\left[\begin{array}{cc}3 & -5 \\ -4 & 2\end{array}\right]$ , (find) $A^{2}-5 A-14 I$ , निकाले , जहाँ I एक इकाई आव्यूह हैं (where I is a unit matrix)
Sol :
$A^{2}-5 A-14 I$
$=\left[\begin{array}{cc}3 & -5 \\ -4 & 2\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ -4 & 2\end{array}\right]-5\left[\begin{array}{cc}3 & -5 \\ -4 & 2\end{array}\right]-14\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$
$=\left[\begin{array}{cc}9+20 & -15-10 \\ -12-8 & 20+4\end{array}\right]-\left[\begin{array}{cc}15 & -25 \\ -20 & 10\end{array}\right]-\left[\begin{array}{cc}14 & 0 \\ 0 & 14\end{array}\right]$
$=\left[\begin{array}{ccc}29 & -25 \\ -20 & \phantom{-} 24\end{array}\right]-\left[\begin{array}{cc}15 & -25 \\ -20 & 10\end{array}\right]-\left[\begin{array}{cc}14 & 0 \\ 0 & 14\end{array}\right]$
$=\left[\begin{array}{cc}14 & 0 \\ 0 & 14\end{array}\right]-\left[\begin{array}{cc}14 & 0 \\ 0 & 14\end{array}\right]$
$=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]=0$
Question 31(i) यदि (If) $A=\left[\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right]$, सत्यापित करें कि (verify that) $A^{2}-7 A-2 I=0$
Sol :
(ii) यदि आव्यूह (If the matrix) $A=\left[\begin{array}{ll}5 & 3 \\ 12 & 7\end{array}\right]$ , जहाँ I एक इकाई आव्यूह है , तो सत्यापित करें, कि (where I is a unit matrix , then verify that)
$A^{2}-12 A-I=O$
Sol :
Question 32
सत्यापित करे कि (Verify that) $A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$ समिकरण (satisfies the equation) $A^{3}-4 A^{2}+A=0$ को संतुष्ट करता है ।
Sol :
$A^{2}=A A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$
$=\left[\begin{array}{cc}4+3 & 6+6 \\ 2+2 & 3+4\end{array}\right]$
$=\left[\begin{array}{ll}7 & 12 \\ 4 & 7\end{array}\right]$
$A^{3}=A^{2} \cdot A=\left[\begin{array}{ll}7 & 12 \\ 4 & 7\end{array}\right]\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$
$=\left[\begin{array}{cc}14+12 & 21+24 \\ 8+7 & 12+14\end{array}\right]$
$=\left[\begin{array}{cc}26 & 45 \\ 15 & 26\end{array}\right]$
L.H.S
$4^{3}-4 A^{2}+A$
$=\left[\begin{array}{cc}26 & 45 \\ 15 & 26\end{array}\right]-4\left[\begin{array}{cc}7 & 12 \\ 4 & 7\end{array}\right]+\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$
$=\left[\begin{array}{cc}28 & 48 \\ 16 & 28\end{array}\right]-\left[\begin{array}{cc}28 & 48 \\ 16 & 28\end{array}\right]$
$=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]=0$
Question 33
(i) यदि (If) $A=\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right]$ तथा (and) $\mathrm{I}_{2}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, k ज्ञात करें ताकि (find k so that ) $A^{2}=8 A+k I_{2}$
Sol :
$\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right]=8\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right]+k\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$
$\left[\begin{array}{cc}1-0 & 0+0 \\ -1-7 & -0+4 y\end{array}\right]=\left[\begin{array}{cc}8 & 0 \\ -8 & 56\end{array}\right]+\left[\begin{array}{cc}k & 0 \\ 0 & k\end{array}\right]$
$\left[\begin{array}{cc}1 & 0 \\ -8 & 4\end{array}\right]=\left[\begin{array}{cc}8+k & 0 \\ -8 & 56+k\end{array}\right]$
⇒1=8+k
⇒-7=k
(ii) यदि (If) $A=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]$, k ज्ञात करें ताकि find k such that $A^{2}=k A-2 I_{2}$
Sol :
(iii) यदि (If) $A=\left[\begin{array}{rr}1 & 0 \\ -1 & 7\end{array}\right]$ , k ज्ञात करें ताकि (find k such that)
$A^{2}-8 A+k I=O$
Sol :
Question 34
(i) यदि (If) $A=\left[\begin{array}{rr}3 & 1 \\ -1 & 2\end{array}\right], f(A)$ निकाले जहाँ (find $f(A)$ , where) $f(x)=x^{2}-5 x+7$
Sol :
$f(A)=A^{2}-5 A+7 I$
$=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]-5\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]+7\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$
$=\left[\begin{array}{cc}9-1 & 3+2 \\ -3-2 & -1+4\end{array}\right]-\left[\begin{array}{cc}15 & 5 \\ -5 & 10\end{array}\right]+\left[\begin{array}{cc}7 & 0 \\ 0 & 7\end{array}\right]$
$=\left[\begin{array}{cc}8 & 5 \\ -5 & 3\end{array}\right]-\left[\begin{array}{cc}15 & 5 \\ -5 & 10\end{array}\right]+\left[\begin{array}{cc}7 & 0 \\ 0 & 7\end{array}\right]$
$=\left[\begin{array}{cc}15 & 5 \\ -5 & 10\end{array}\right]-\left[\begin{array}{cc}15 & 5 \\ -5 & 10\end{array}\right]$
$=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]=0$
(ii) यदि (If) $A=\left[\begin{array}{rr}3 & 4 \\ -4 & -3\end{array}\right]$, f(A) निकालें जहाँ (find f(A), where) $f(x)=x^{2}-5 x+7$
Sol :
(iii) $A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right], f(x)=x^{2}-2 x-3$ दिखाएँ कि (show that) f(A)=0
Sol :
Question 35
(i) यदि (if) $A=\left[\begin{array}{ll}2 & -1 \\ 3 & 2\end{array}\right]$ तथा (and) $B=\left[\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right]$ (find) $\left(3 A^{2}-2 B\right)$ निकाले ।
Sol :
$3 A^{2}-2 B=3\left[\begin{array}{cc}2 & -1 \\ 3 & 2\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 3 & 2\end{array}\right]-2\left[\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right]$
$=3\left[\begin{array}{cc}4-3 & -2-2 \\ 6+6 & -3+4\end{array}\right]-\left[\begin{array}{cc}0 & 8 \\ -2 & 14\end{array}\right]$
$=3\left[\begin{array}{rr}1 & -4 \\ 12 & 1\end{array}\right]-\left[\begin{array}{cc}0 & 8 \\ -2 & 14\end{array}\right]$
$=\left[\begin{array}{rr}3 & -12 \\ 36 & 3\end{array}\right]-\left[\begin{array}{cc}0 & 8 \\ -2 & 14\end{array}\right]$
$=\left[\begin{array}{cc}3 & -20 \\ 38 & -11\end{array}\right]$
(ii) यदि (If)$A=\left[\begin{array}{cc}2 & -1 \\ 3 & 2\end{array}\right] $, $ B=\left[\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right]$ ( find ) $3 A^{2}-2 B+I$ निकाले ।
Sol :
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