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https://beta.mathcity.org/_media/logo.pngtext/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 1, Exercise 2.1
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Question 1, Exercise 2.1
Solutions of Question 1 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{lll}1 & 3 & 0 \\ 2 & 0 & 1\end{array}\right]$\begin{align}\text{Order of A}&= 2\times 3\end{align}$B=\left[\begin{array}{ll}1 & 2 \\ 2 & 3 \\ 3 & 4\end{array}\right]$\begin{align}\text{Order of B}&= 3\times 2\end{align}$C…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 2, Exercise 2.1
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Question 2, Exercise 2.1
Solutions of Question 2 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\quad A=\left[\begin{array}{lll}3 & 6 & 2 \\ 2 & 1 & 9\end{array}\right]$$B=\left[\begin{array}{ll}\frac{1}{3} & 1 \\ 2 & 6\end{array}\right]$$C=\left[\begin{array}{l}3 \\ 2 \\ 8\end{array}\right]$$D=\left[\begin{array}{lll}1 & 6 & 9 \\ 2 & 0 …text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 3, Exercise 2.1
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Question 3, Exercise 2.1
Solutions of Question 3 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$A=\begin{bmatrix}
3 & 0 & 0 \\
0 & 1 & 0 \\
2 & 6 & 0
\end{bmatrix}$$$$B=\begin{bmatrix}
-6 & 0 & 0 \\
0 & -6 & 0 \\
0 & 0 & -6
\end{bmatrix}$$$$C=\begin{bmatrix}
1 & 0 \\
2 & 0
\end{bmatrix}$$$$D=\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 &…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 4, Exercise 2.1
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Question 4, Exercise 2.1
Solutions of Question 4 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $$
A=\left[\begin{array}{ccc}
2 & 0 \\
\sqrt{5} & 6 \\
1 & 9
\end{array}\right]$$$$
A^t=\begin{bmatrix}
2 & \sqrt{5} & 1 \\
0 & 6 & 9
\end{bmatrix}$$$$B=\left[\begin{array}{cccc}
1 & 6 & 2 & 0
\end{array}\right] $$$$B^t=\left[\begin{array}{c}
1…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 1, Exercise 2.2
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Question 1, Exercise 2.2
Solutions of Question 1 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[a_{i j}\right]$$2 \times 2$$a_{i j}=\dfrac{i+3 j}{2}$\( a_{ij} = \dfrac{i + 3j}{2} \)\( i = 1, j = 1 \)\[
a_{11} = \dfrac{1 + 3 \cdot 1}{2} = \dfrac{1 + 3}{2} = \dfrac{4}{2} = 2
\]\( i = 1, j = 2 \)\[
a_{12} = \dfrac{1 + 3 \cdot 2}{2} …text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 3, Exercise 2.2
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Question 3, Exercise 2.2
Solutions of Question 3 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & -1 & 2 \\ 0 & 6 & 1 \\ -1 & 0 & -3\end{array}\right]$$B=\left[\begin{array}{ccc}2 & 1 & 7 \\ 0 & 2 & -1 \\ -3 & 4 & 2\end{array}\right]$$C$$A+B+C=0$$$A+B+C=0,$$$$C=-A-B.$$\begin{align*}
C&=-\begin{bmatrix}3 & -1 &…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 3, Exercise 2.2
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Question 3, Exercise 2.2
Solutions of Question 3 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & -1 & 2 \\ 0 & 6 & 1 \\ -1 & 0 & -3\end{array}\right]$$B=\left[\begin{array}{ccc}2 & 1 & 7 \\ 0 & 2 & -1 \\ -3 & 4 & 2\end{array}\right]$$C$$A+B+C=0$$$A+B+C=0,$$$$C=-A-B.$$\begin{align*}
C&=-\begin{bmatrix}3 & -1 &…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 4, Exercise 2.2
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Question 4, Exercise 2.2
Solutions of Question 4 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$\begin{align}\left[\begin{array}{cc} 2 & 1 \\ 3 & 2 \end{array}\right]A\left[\begin{array}{cc} 1 & 3 \\ 2 & 4 \end{array}\right]&=\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]\end{align}$ B = \left[\begin{array}{cc} 2 & 1 \\ 3…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 5, Exercise 2.2
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Question 5, Exercise 2.2
Solutions of Question 5 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $X=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$$X^{2}-4 X-5 I=0$\begin{align}L.H.S. & =X^{2}-4 X-5 I \\
&=\begin{bmatrix}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 2 & 2 \\
2 & 1 & 2…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 6, Exercise 2.2
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Question 6, Exercise 2.2
Solutions of Question 6 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{cc}2 & 1 \\ 3 & -3\end{array}\right]$$\alpha$$\beta$$A^{2}+\alpha I=\beta A$\begin{align}
& A^{2}+\alpha I=\beta A\\
\implies &\begin{bmatrix}
2 & 1 \\
3 & -3
\end{bmatrix}
\begin{bmatrix}
2 & 1 \\
3 & -3
\end{bmatrix}+\a…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 7, Exercise 2.2
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Question 7, Exercise 2.2
Solutions of Question 7 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ll}x & 0 \\ y & 1\end{array}\right]$$n, A^{n}=\left[\begin{array}{cc}x^{n} & 0 \\ \dfrac{y\left(x^{n}-1\right)}{x-1} & 1\end{array}\right]$$$A = \begin{bmatrix} x & 0 \\ y & 1 \end{bmatrix}.$$$n = 1$\begin{align}A^1 =\beg…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 8, Exercise 2.2
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Question 8, Exercise 2.2
Solutions of Question 8 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$B$$2 \times 3$$3 \times 2$$(A B)^{t}=B^{t} A^{t}$\( A \)\( B \)\( 2 \times 3 \)\( 3 \times 2 \)\begin{align*}
A &= \begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23}
\end{bmatrix}\\
B &= \begin{bmatrix}
b_{11} & b_{12}…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 9, Exercise 2.2
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Question 9, Exercise 2.2
Solutions of Question 9 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$B$$3 \times 3$$(A+B)^{t}=A^{t}+B^{t}$\begin{align*}
A &= \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix} \\
B &= \begin{pmatrix}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 10, Exercise 2.2
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Question 10, Exercise 2.2
Solutions of Question 10 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$B$$A B=B$$B A=A$$A^{2}+B^{2}$$$AB = B$$$$BA = A$$\begin{align*}
A^2 &= AA\\
& = A(BA)\\
&=(AB)A\\
&=BA\\
&=A
\end{align*}\begin{align*}
B^2&= BB \\
&=B(AB)\\
& = (BA)B\\
&=AB\\
&=B\end{align*}$$A^2 + B^2 = A + B$$$AB = B$$BA = A$$$A^2 + B…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 11, Exercise 2.2
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Question 11, Exercise 2.2
Solutions of Question 11 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[a_{i j}\right]$$3 \times 3$$a_{i j}=i^{2}-j^{2}$$A$$A=\left[a_{i j}\right]$$a_{ij}=a+{ji}$$a_{ij}=-a_{ji}$$a_{i j}=i^{2}-j^{2}$\begin{align}
a_{ji} & = j^2 -i^2 \\
&= - (i^2 -j^2) \\
&= - a_{ij}
\end{align}$a_{ij}=-a_{ji}$text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 12, Exercise 2.2
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Question 12, Exercise 2.2
Solutions of Question 12 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$\left(A^{n}\right)^{t}=\left(A^{t}\right)^{n}$text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 13, Exercise 2.2
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Question 13, Exercise 2.2
Solutions of Question 13 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $X$$Y$$2 X-Y=\left[\begin{array}{ccc}1 & 6 & -3 \\ 2 & 1 & 7\end{array}\right]$$X+3 Y=\left[\begin{array}{ccc}4 & 3 & 2 \\ 1 & -3 & 0\end{array}\right]$\begin{align*}
2X - Y = \begin{pmatrix} 1 & 6 & -3 \\ 2 & 1 & 7 \end{pmatrix} \cdots (i)\\…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 1, Exercise 2.3
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Question 1, Exercise 2.3
Solutions of Question 1 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]$\begin{align*}
A &= \left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]\\
|A|&=2(-2-2)-3(2-8)+1(1+4)\\
\implies |A|&=-8+18+5\\
\implies |…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 2, Exercise 2.3
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Question 2, Exercise 2.3
Solutions of Question 2 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{lll}3 & 2 & 3 \\ 4 & 5 & 1 \\ 2 & 1 & 0\end{array}\right]$\(R_1\)\(a_{11} = 3\)\(a_{12} = 2\)\(a_{13} = 3\)\begin{align*}
A &= \left[\begin{array}{ccc} 3 & 2 & 3 \\ 4 & 5 & 1 \\ 2 & 1 & 0 \end{array}\right]\\
& A_{11} = (-1…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 3, Exercise 2.3
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Question 3, Exercise 2.3
Solutions of Question 3 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}3 & 1 & 2 \\ 2 & 3 & 1 \\ -4 & 1 & -3\end{array}\right]$\begin{align*}
A &= \left[\begin{array}{ccc} 3 & 1 & 2 \\ 2 & 3 & 1 \\ -4 & 1 & -3\end{array}\right]\end{align*}\(3 \times 3\)\begin{align*}
|A| &= 3(3 \cdot (-3) …text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 4, Exercise 2.3
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Question 4, Exercise 2.3
Solutions of Question 4 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\lambda$$\left[\begin{array}{lll}\lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1\end{array}\right]$\begin{align*}
A &= \left[\begin{array}{ccc}
\lambda & 1 & 3 \\
2 & 1 & 8 \\
0 & 3 & 1
\end{array}\right]\\
|A| &= \lambda \cdot (-23) - 1 \cdot 2 + 3…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 5, Exercise 2.3
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Question 5, Exercise 2.3
Solutions of Question 5 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1\end{array}\right]$\begin{align*}
A &= \left[\begin{array}{ccc}
1 & -1 & 1 \\
2 & 1 & -1 \\
1 & -2 & -1
\end{array}\right]\\
|A|&= 1 [-1 - 2] + 1 [-2 + 1] + 1 [-4 - 1] \\
&= 1 \cd…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 6, Exercise 2.3
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Question 6, Exercise 2.3
Solutions of Question 6 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}2 & 1 & -3 \\ 0 & 1 & 0 \\ 2 & 1 & 6\end{array}\right]$$A^{-1}$$A A^{-1}=A^{-1} A=I_{3}$\begin{align*} A &= \begin{bmatrix}
2 & 1 & -3 \\
0 & 1 & 0 \\
2 & 1 & 6
\end{bmatrix} \end{align*}$ A^{-1} $$ A $\begin{align*}
…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 7, Exercise 2.3
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Question 7, Exercise 2.3
Solutions of Question 7 of Exercise 2.3 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $(A B)^{-1}=B^{-1} A^{-1}$$A=\left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right]$$B=\left[\begin{array}{ll}3 & 2 \\ 0 & 2\end{array}\right]$\begin{align*}
A &= \left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right] \\
|A|& = 12 - 8 = 4\\ …text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 1, Exercise 2.5
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Question 1, Exercise 2.5
Solutions of Question 1 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{array}\right]$\begin{align*}
& \quad \left[\begin{array}{ccc}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{array}\right]\\
\sim & \text{R}
\left[\begin{array}{ccc}
1 & 3 & 5 \\
0 & …text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 2, Exercise 2.5
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Question 2, Exercise 2.5
Solutions of Question 2 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}5 & 9 & 3 \\ 3 & -5 & 6 \\ 2 & 10 & 6\end{array}\right]$\begin{align*}&\quad\left[ \begin{array}{ccc}
5 & 9 & 3 \\
3 & -5 & 6 \\
2 & 10 & 6
\end{array} \right]\\
\sim & \text{R}\left[ \begin{array}{ccc}
1 & \frac{9}{…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 3, Exercise 2.5
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Question 3, Exercise 2.5
Solutions of Question 3 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left[\begin{array}{ccc}0 & -1 & -1 \\ -1 & 3 & 0 \\ 1 & -1 & 4\end{array}\right]$$A A^{-1}=A^{-1} A=I$\begin{align*}
A&=\left[ \begin{array}{ccc}
0 & -1 & -1 \\
-1 & 3 & 0 \\
1 & -1 & 4
\end{array} \right]\\
|A|&=0+1(-4)-1(1-3)\\
&=-4+3\…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 1, Exercise 2.6
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Question 1, Exercise 2.6
Solutions of Question 1 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $ 2 x_{1}-3 x_{2}+4 x_{3}=0$$x_{1}-2 x_{2}+3 x_{3}=0$$4 x_{1}+x_{2}-6 x_{3}=0$\begin{align*}
&2 x_{1}-3 x_{2}+4 x_{3}=0\cdots (i)\\
&x_{1}-2 x_{2}+3 x_{3}=0\cdots (ii)\\
&4 x_{1}+x_{2}-6 x_{3}=0\cdots (iii)\\
\end{align*}\begin{align*}
A &= \le…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 2, Exercise 2.6
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Question 2, Exercise 2.6
Solutions of Question 2 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\lambda$$\lambda$$2 x_{1}-\lambda x_{2}+x_{3}=0$$2 x_{1}+3 x_{2}-x_{3}=0$$3 x_{1}-2 x_{2}+4 x_{3}=0$\begin{align*}
&2 x_{1}-\lambda x_{2}+x_{3}=0 \cdots(i)\\
&2 x_{1}+3 x_{2}-x_{3}=0\cdots(ii)\\
&3 x_{1}-2 x_{2}+4 x_{3}=0\cdots(iii)\\
\end{ali…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 3, Exercise 2.6
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Question 3, Exercise 2.6
Solutions of Question 3 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x+3 y+4 z=2$$2 x+y+z=5$$3 x-2 y+z=-3$\begin{align*}
\begin{aligned}
2x + 3y + 4z &= 2 \\
2x + y + z &= 5 \\
3x - 2y + z &= -3
\end{aligned}\end{align*}\begin{align*}
A_{b} &=\quad \left[\begin{array}{cccc}
2 & 3 & 4 & 2 \\
2 & 1 & 1 & 5 \\
3…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 4, Exercise 2.6
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Question 4, Exercise 2.6
Solutions of Question 4 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x_{1}-x_{2}-x_{3}=2$$3 x_{1}-4 x_{2}+3 x_{3}=7$$4 x_{1}+2 x_{2}-5 x_{3}=10$\begin{align*}
2x_1 - x_2 - x_3 &= 2, \\
3x_1 - 4x_2 + 3x_3 &= 7, \\
4x_1 + 2x_2 - 5x_3 &= 10,
\end{align*}\begin{align*}
A_b &= \begin{bmatrix}
2 & -1 & -1 & : & 2 …text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 5, Exercise 2.6
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Question 5, Exercise 2.6
Solutions of Question 5 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $x_{1}+x_{2}+2 x_{3}=8$$-x_{1}-2 x_{2}+3 x_{3}=1$$3 x_{1}-7 x_{2}+4 x_{3}=10$$A X=B$\begin{align*}
&A = \begin{bmatrix}
1 & 1 & 2 \\
-1 & -2 & 3 \\
3 & -7 & 4
\end{bmatrix}, \quad
X = \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}, \quad
B = \…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 6, Exercise 2.6
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Question 6, Exercise 2.6
Solutions of Question 6 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $5 x+3 y+z=6$$2 x+y+3 z=19$$x+2 y+4 z=25$\begin{align*}
A &= \begin{bmatrix}
5 & 3 & 1 \\
2 & 1 & 3 \\
1 & 2 & 4
\end{bmatrix}, \quad
X = \begin{bmatrix}
x \\
y \\
z
\end{bmatrix}, \quad
B = \begin{bmatrix}
6 \\
19 \\
25
\end{bmatrix}
\end{alig…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 7 and 8, Exercise 2.6
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Question 7 and 8, Exercise 2.6
Solutions of Question 7 and 8 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}3 & 2 & 1 \\ 4 & -1 & 2 \\ 7 & 3 & -3\end{array}\right]$$A^{-1}$$3 x+4 y+7 z=14 ; 2 x-y+3 z=4 ; \quad x+2 y-3 z=0$\begin{align*}
A &= \begin{bmatrix}
3 & 2 & 1 \\
4 & -1 & 2 \\
7 & 3 & -3
\end{bmatrix}\\
|…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 9 and 10, Exercise 2.6
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Question 9 and 10, Exercise 2.6
Solutions of Question 9 and 10 of Exercise 2.6 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $2 x-y+3 z=\alpha ; 3 x+y-5 z=\beta ;-5 x-5 y+21 z=\gamma$$\gamma \neq 2 \alpha-3 \beta$$2$$2$$3$$3$text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 1, Review Exercise
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Question 1, Review Exercise
Solutions of Question 1 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A$$m \times n$$B$$n \times p$$A B$$n \times p$$m \times p$$p \times m$$n \times n$$A$$1 \times n$$A^{t} A$$1 \times n$$n \times 1$$1 \times 1$$n \times n$$a_{i j}$$A$$a_{i j}=(-1)^{i+j} A_{i j}$$a_{i j}=(-1)^{i+j} M_{i j}$$\frac{A_{i j}}…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 2 and 3, Review Exercise
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Question 2 and 3, Review Exercise
Solutions of Question 2 and 3 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $A=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]$$A_{13}, A_{23}$$A_{33}$$|A|$\begin{align*}
A&=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]\\
A_{13} &= (-1)^{…text/html2025-01-21T16:13:59+00:00Anonymous (anonymous@undisclosed.example.com)Question 4 and 5, Review Exercise
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Question 4 and 5, Review Exercise
Solutions of Question 4 and 5 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. $\left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|=(a+1+2 l)(a+1-l)^{2}$\begin{align*}
L.H.S &= \left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|\\
&=\left|\b…