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Question 18, Exercise 2.2
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ingular matrix.\\ $$A=\left[ \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} \right]$$ $$|A|=a_{11}a_{22}-a_{12}a_{21}$$ $$AdjA=\left[ \begin{matrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \\ \end{matrix} \right]$$ $$A^{-1}=\dfrac{1}{|A|}AdjA$$ $$A^{-1}=\dfrac{1}{a_{11}}a_{22}-a_{12}a_{21}\left[ \begin{matrix} a_{2
Question 13, Exercise 2.1
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====Solution==== $$A=\left[ \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23... trix} \right]$$ $$A^t=\left[ \begin{matrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32... t=( A+A^t )$$ $$A+A^t=\left[ \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{2... \end{matrix} \right]+\left[ \begin{matrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32
Question 3, Exercise 2.2
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$. ====Solution==== Let $$A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23... 33} \\ \end{bmatrix}$$ Then \begin{align}|A|&=a_{11} \left( a_{22} a_{33}-a_{23} a_{32} \right)-a_{12... 3}\left( a_{21}a_{32}-a_{22}a_{31} \right)\\ &=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}
Question 11, Exercise 2.1
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====== Question 11, Exercise 2.1 ====== Solutions of Question 11 of Exercise 2.1 of Unit 02: Matrices and Determinants. ... KPTB or KPTBB) Peshawar, Pakistan. =====Question 11===== Let $A=\begin{bmatrix}0 & 1 & -2 \\-1 & 0 &... & 0 \end{bmatrix}$ and $B=\begin{bmatrix}0 & -6 & 11 \\6 & 0 & -7 \\-11 & 7 & 0 \end{bmatrix}$. Veri
Question 3, Exercise 2.1
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\right]\\ \implies A( B+C )&=\left[ \begin{matrix}11 & 15 & 8 \\3 & \,\,20 & \,6 \\\end{matrix}\righ... left[ \begin{matrix}\quad 2 & 13 & 3 \\-2 & 15 & 11 \\\end{matrix} \right].\end{align} and \begin{a... AC &=\left[ \begin{matrix}2 & 13 & 3 \\-2 & 15 & 11 \\ \end{matrix} \right]+\left[ \begin{matrix}9 ... begin{matrix}\quad2+9 & 13+2 & 3+5\\-2+5 & 15+5 & 11-5\\ \end{matrix} \right] \\ \implies AB+AC&=\lef
Question 4, Exercise 2.1
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\end{matrix} \right]\\ &=\left[ \begin{matrix} 11 & 8 & 8 \\ 8 & 11 & 8 \\ 8 & 8 & 11 \\ \end{matrix} \right]\end{align} Now we take $$2A=\left[ \begin{matrix} 2... frac{1}{3}A^2-2A-9I \\ =&\left[ \begin{matrix} 11 & 8 & 8 \\ 8 & 11 & 8 \\ 8 & 8 & 11 \\ \
Question 10, Exercise 2.1
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\right]$$ $$A+B=\left[ \begin{matrix} 6 & 3 & 11 \\ 3 & -6 & -2 \\ 11 & -2 & 1 \\ \end{matrix} \right]$$ $$( A+B)^t=\left[ \begin{matrix} 6 & 3 & 11 \\ 3 & -6 & -2 \\ 11 & -2 & 1 \\ \end{matrix} \right]$$ $$A^t+B^t=\left[ \begin{matrix} 1
Question 16 & 17, Exercise 2.2
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=\left[ \begin{matrix} \dfrac{1}{15} & \dfrac{-11}{15} \\ \dfrac{1}{15} & \dfrac{4}{15} \\ \en... trix} \right]$$ $$AB=\left[ \begin{matrix} 4 & 11 \\ -1 & 1 \\ \end{matrix} \right]$$ $$\Rightarrow |AB|=4+11$$ $$\Rightarrow \,\,|AB|=15$$ $$AdjAB=\left[ \begin{matrix} 1 & -11 \\ 1 & 4 \\ \end{matrix} \right]$$ $$( AB )^
Question 11, Exercise 2.2
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====== Question 11, Exercise 2.2 ====== Solutions of Question 11 of Exercise 2.2 of Unit 02: Matrices and Determinants. ... KPTB or KPTBB) Peshawar, Pakistan. =====Question 11(i)===== Identify singular and non-singular matric... 6-12$$ $$|A|=0$$ $A$ is singular. =====Question 11(ii)===== Identify singular and non-singular matri
Question 1, Exercise 2.2
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& 0 \\2 & 0 & -2 \end{bmatrix}$ , then find $A_{11},A_{21},A_{23},A_{31},A_{32},A_{33}.$ Also find $... 2 & 0 & -2 \\ \end{matrix} \right]$$ $${{A}_{11}}={{\left( -1 \right)}^{1+1}}\left| \begin{matrix... 0 & -2 \\ \end{matrix} \right|$$ $$\implies{{A}_{11}}=-4$$ $${{A}_{21}}={{\left( -1 \right)}^{2+1}}\l... implies{{A}_{33}}=5$$ Now \begin{align}|A|&={{a}_{11}}{{A}_{11}}+{{a}_{12}}{{A}_{12}}+{{a}_{13}}{{A}_{
Question 2, Exercise 2.3
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im}{R}&\left[\begin{matrix} 1 & 4 & 14 \\ 0 & 1 & 11 \\ 0 & 6 & 17 \end{matrix}\left| \begin{matrix} 1... }{R}&\left[\begin{matrix} 1 & 0 & -30 \\ 0 & 1 & 11 \\ 0 & 0 & -49 \end{matrix}\left| \begin{matrix}... m}{R}&\left[\begin{matrix} 1 & 0 & -30 \\ 0 & 1 & 11 \\ 0 & 0 & 1 \end{matrix}\left| \begin{matrix} -... ight. \right]\text{ by }R_1+30R_3\text{ and } R_2-11R_3\end{align} Thus we have \begin{align} A^{-1}&=
Question 14 & 15, Exercise 2.2
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}{|A|}$$ $$Adj\,\,A={{\left[ \begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23... } & A_{33} \\ \end{matrix} \right]}^{t}}$$ $$A_{11}=(-1)^{1+1}\left| \begin{matrix} 3 & 2 \\ 0 & 5 \\ \end{matrix} \right|$$ $$A_{11}=15$$ $$A_{12}=(-1)^{1+2}\left| \begin{matrix} ... === <text align="left"><btn type="primary">[[math-11-kpk:sol:unit02:ex2-2-p11 |< Question 13]]</btn></
Question 4, Exercise 2.3
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& 5 \\3 & 4 & 5 & 6 \\4 & 5 & 6 & 7 \\9 & 10 & 11 & 12\end{bmatrix}$ ====Solution==== \begin{align}... \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix}\\ \underset{\sim}{R}&\begin{bm... \\ 1 & 1 & 1 & 1 \\ 4 & 5 & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix} \text{by}R_2-R_1\\ \underset{\... \\ 2 & 3 & 4 & 5 \\ 4 & 5 & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix} \text{by}R_1\leftrightarrow R_
Question 5 & 6, Exercise 2.1
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{matrix} \right]\\ &=\left[ \begin{matrix}7 & 2 & 11 \\0 & 4 & 11 \\ \end{matrix} \right]\end{align} =====Question 6(ii)===== Solve the matrix equati... === <text align="left"><btn type="primary">[[math-11-kpk:sol:unit02:ex2-1-p4 |< Question 4]]</btn></te... t> <text align="right"><btn type="success">[[math-11-kpk:sol:unit02:ex2-1-p6|Question 7 >]]</btn></tex
Question 12, Exercise 2.1
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=== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit02:ex2-1-p10 |< Question 11]]</btn></text> <text align="right"><btn type="success">[[math-11-kpk:sol:unit02:ex2-1-p12|Question 13 >]]</btn></t
Question 4, Exercise 2.2
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Question 8,9 & 10, Exercise 2.2
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Question 12, Exercise 2.2
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Question 2, Exercise 2.1
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Question 7, Exercise 2.1
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Question 8, Exercise 2.1
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Question 9, Exercise 2.1
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Question 2, Exercise 2.2
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Question 5, Exercise 2.2
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Question 6, Exercise 2.2
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Question 7, Exercise 2.2
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Question 13, Exercise 2.2
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Question 1, Exercise 2.3
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Question 19, Exercise 2.2
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