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- Question 2, Exercise 2.3
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... Question 2(i)===== Find the inverse of the matrix by using elementary row operation. $$\begin{bmatrix}... \\ 0 & 0 & 1 \end{matrix} \right. \right]\text{ by }R_1+3R_3\text{ and }R_2+2R_3 \\ \underset{\sim}{... \\ 1 & 0 & 4 \end{matrix} \right. \right]\text{ by }R_3+R_1\\ \underset{\sim}{R}&\left[\begin{matrix
- Question 1, Exercise 2.3
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... \ 0 & -5 & 6 \\ 0 & -5 & -2 \end{bmatrix} \text{ by } R_2-2R_1 \text{ and } R_3-3R_1 \\ \underset{\si... 0 & -5 & 6 \\ 0 & -5 & -2 \end{bmatrix} \text{ by } R_3-R_2 \\ \underset{\sim}{R}&\begin{bmatrix} 1... \\ 3 & \quad 1 & 3 & \quad 2 \end{bmatrix}\text{ by } R_1\leftrightarrow R_2\\ \underset{\sim}{R}& \b
- Question 3, Exercise 2.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... & c\end{bmatrix}\\ &=\begin{bmatrix}ax+hy+gz & hx+by+fz & gx+fy+cz \end{bmatrix} \\ &=\begin{bmatrix}ax+hy+gz & hx+by+fz & gx+fy+cz \end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}\\ &=\left[ x( ax+hy+gz )+y( hx+by+fz )+z( gx+fy+cz ) \right] \\ &=\left[ a{{x}^{2}}
- Question 5, Exercise 2.2
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... b+3b & 3-3c+3c \\ 4 & 5 & 6 \end{vmatrix} \text{ by } R_2-3R_1 \\ &=\begin{vmatrix} a & b & c \\1 & ... & c\\ a+b+c & b+c+a & c+a+b \end{vmatrix} \text{ by } R_3+R_2 \\ &=(a+b+c)\begin{vmatrix} 1 & 1 & 1\\... text{taking common from }R_3 \\ &=(a+b+c)0 \text{ by } R_1\simeq R_3 \\ &=0=R.H.S.\end{align} =====
- Question 3, Exercise 2.3
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... \\ 0 & 2 & 5 \\ 0 & 2 & 1 \end{bmatrix} \text{ by }R_2-2R_1 \text{ and } R_1-2R_3\\ \underset{\sim}... 2 \\ 0 & 0 & 4 \\ 0 & 2 & 1 \end{bmatrix}\text{ by }R_2-R_3\end{align} The last matrix is the echelo... \\ 0 & 2 & 1 \\ 3 & 1 & -4 \end{bmatrix} \text{ by }R_1\leftrightarrow R_3\\ \underset{\sim}{R}&\beg
- Question 4, Exercise 2.3
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix} \text{by}R_2-R_1\\ \underset{\sim}{R}&\begin{bmatrix} 1 & ... & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix} \text{by}R_1\leftrightarrow R_2\\ \underset{\sim}{R}&\begi... & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 \end{bmatrix} \text{by}R_2-2R_1\text{,}R_3-4R_1 \text{and} R_4-9R_1\\ \u
- Question 6, Exercise 2.2
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... {c} \\ \end{matrix} \right|$$ Multiply first row by $a$, second row by $b$ and third row by $c.$ We have, $$=\dfrac{1}{abc}\left| \begin{matrix} abc & a^4 & a\dfrac{1}{a
- Question 2, Exercise 2.2
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... \2 & 4 & 2\\ \end{matrix} \right|$$ Multiply $-3$ by third column of R.H.S. =====Question 2(v)===== ... \\ \end{matrix} \right|$$ Multiply the third row by $2$ and add it in 2nd row of L.H.S to get R.H.S.
- Question 16 & 17, Exercise 2.2
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)... 3}{50}+\dfrac{2}{50}$$ $$|A^{-1}|=\dfrac{1}{10}$$ By using (1), above expression gives,\\ $$|A^{-1}|=\
- Question 1, Exercise 2.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)
- Question 2, Exercise 2.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)
- Question 4, Exercise 2.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)
- Question 5 & 6, Exercise 2.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)
- Question 7, Exercise 2.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)
- Question 8, Exercise 2.1
- Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB)