Determinants

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ Let\Delta =\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill \left(1+sin\theta \right)\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1+cos\theta \hfill \end{array}\right|\\ C_1\to C_1-C_2,C_2\to C_2-C_3\\ =\left|\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill -sin\theta \hfill & \hfill sin\theta \hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill -cos\theta \hfill & \hfill 1+cos\theta \hfill \end{array}\right|\\ Expandingalong{C}_3\\ =1\left|\begin{array}{cc}\hfill -sin\theta \hfill & \hfill sin\theta \hfill \\ \hfill 0\hfill & \hfill -cos\theta \hfill \end{array}\right|=sin\theta cos\theta -0=sin\theta cos\theta \\ =\frac{1}{2}.2sin\theta cos\theta =\frac{1}{2}sin2\theta \\ =\frac{1}{2}*1=\frac{1}{2}\left[Maximumvalueofsin2\theta =1\right]\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ Giventhat\Delta =\left|\begin{array}{ccc}\hfill a\hfill & \hfill p\hfill & \hfill x\hfill \\ \hfill b\hfill & \hfill q\hfill & \hfill y\hfill \\ \hfill c\hfill & \hfill r\hfill & \hfill z\hfill \end{array}\right|=16\\ L.H.S.{\Delta }_1=\left|\begin{array}{ccc}\hfill p+x\hfill & \hfill a+x\hfill & \hfill a+p\hfill \\ \hfill q+y\hfill & \hfill b+y\hfill & \hfill b+q\hfill \\ \hfill r+z\hfill & \hfill c+z\hfill & \hfill c+r\hfill \end{array}\right|\\ C_1\to C_1+C_2+C_3\\ =\left|\begin{array}{ccc}\hfill 2p+2x+2a\hfill & \hfill a+x\hfill & \hfill a+p\hfill \\ \hfill 2q+2y+2b\hfill & \hfill b+y\hfill & \hfill b+q\hfill \\ \hfill 2r+2z+2c\hfill & \hfill c+z\hfill & \hfill c+r\hfill \end{array}\right|\\ =2\left|\begin{array}{ccc}\hfill p+x+a\hfill & \hfill a+x\hfill & \hfill a+p\hfill \\ \hfill q+y+b\hfill & \hfill b+y\hfill & \hfill b+q\hfill \\ \hfill r+z+c\hfill & \hfill c+z\hfill & \hfill c+r\hfill \end{array}\right|\left[Taking2commonfrom{C}_1\right]\\ C_1\to C_1-C_2\\ =2\left|\begin{array}{ccc}\hfill p\hfill & \hfill a+x\hfill & \hfill a+p\hfill \\ \hfill q\hfill & \hfill b+y\hfill & \hfill b+q\hfill \\ \hfill r\hfill & \hfill c+z\hfill & \hfill c+r\hfill \end{array}\right|\\ C_3\to C_3-C_1\\ =2\left|\begin{array}{ccc}\hfill p\hfill & \hfill a+x\hfill & \hfill a\hfill \\ \hfill q\hfill & \hfill b+y\hfill & \hfill b\hfill \\ \hfill r\hfill & \hfill c+z\hfill & \hfill c\hfill \end{array}\right|\\ Splittingup{C}_2\\ =2\left|\begin{array}{ccc}\hfill p\hfill & \hfill a\hfill & \hfill a\hfill \\ \hfill q\hfill & \hfill b\hfill & \hfill b\hfill \\ \hfill r\hfill & \hfill c\hfill & \hfill c\hfill \end{array}\right|+2\left|\begin{array}{ccc}\hfill p\hfill & \hfill x\hfill & \hfill a\hfill \\ \hfill q\hfill & \hfill y\hfill & \hfill b\hfill \\ \hfill r\hfill & \hfill z\hfill & \hfill c\hfill \end{array}\right|=2\left(0\right)+2\left|\begin{array}{ccc}\hfill p\hfill & \hfill x\hfill & \hfill a\hfill \\ \hfill q\hfill & \hfill y\hfill & \hfill b\hfill \\ \hfill r\hfill & \hfill z\hfill & \hfill c\hfill \end{array}\right|\\ =2\left|\begin{array}{ccc}\hfill p\hfill & \hfill x\hfill & \hfill a\hfill \\ \hfill q\hfill & \hfill y\hfill & \hfill b\hfill \\ \hfill r\hfill & \hfill z\hfill & \hfill c\hfill \end{array}\right|\Rightarrow 2\left|\begin{array}{ccc}\hfill a\hfill & \hfill p\hfill & \hfill x\hfill \\ \hfill b\hfill & \hfill q\hfill & \hfill y\hfill \\ \hfill c\hfill & \hfill r\hfill & \hfill z\hfill \end{array}\right|\left(C_1\leftrightarrow C_{3}and{C}_2\leftrightarrow C_3\right)\\ =2*16=32\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ Let\Delta =\left|\begin{array}{ccc}\hfill x+a\hfill & \hfill p+u\hfill & \hfill l+f\hfill \\ \hfill y+b\hfill & \hfill q+v\hfill & \hfill m+g\hfill \\ \hfill z+c\hfill & \hfill r+w\hfill & \hfill n+h\hfill \end{array}\right|\\ Splittingup{C}_1\\ =\left|\begin{array}{ccc}\hfill x\hfill & \hfill p+u\hfill & \hfill l+f\hfill \\ \hfill y\hfill & \hfill q+v\hfill & \hfill m+g\hfill \\ \hfill z\hfill & \hfill r+w\hfill & \hfill n+h\hfill \end{array}\right|+\left|\begin{array}{ccc}\hfill a\hfill & \hfill p+u\hfill & \hfill l+f\hfill \\ \hfill b\hfill & \hfill q+v\hfill & \hfill m+g\hfill \\ \hfill c\hfill & \hfill r+w\hfill & \hfill n+h\hfill \end{array}\right|\\ Splittingup{C}_{2}inboth\text{\hspace{0.17em}determinants}\\ =\left|\begin{array}{ccc}\hfill x\hfill & \hfill p\hfill & \hfill l+f\hfill \\ \hfill y\hfill & \hfill q\hfill & \hfill m+g\hfill \\ \hfill z\hfill & \hfill r\hfill & \hfill n+h\hfill \end{array}\right|+\left|\begin{array}{ccc}\hfill x\hfill & \hfill u\hfill & \hfill l+f\hfill \\ \hfill y\hfill & \hfill v\hfill & \hfill m+g\hfill \\ \hfill z\hfill & \hfill w\hfill & \hfill n+h\hfill \end{array}\right|+\left|\begin{array}{ccc}\hfill a\hfill & \hfill p\hfill & \hfill l+f\hfill \\ \hfill b\hfill & \hfill q\hfill & \hfill m+g\hfill \\ \hfill c\hfill & \hfill r\hfill & \hfill n+h\hfill \end{array}\right|+\left|\begin{array}{ccc}\hfill a\hfill & \hfill u\hfill & \hfill l+f\hfill \\ \hfill b\hfill & \hfill v\hfill & \hfill m+g\hfill \\ \hfill c\hfill & \hfill w\hfill & \hfill n+h\hfill \end{array}\right|\\ Similarlybysplitting{C}_{3}ineach\text{\hspace{0.17em}determinants},wewillget\mathrm{8determinants}.\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ Let\Delta =\left|\begin{array}{ccc}\hfill sinA\hfill & \hfill cosA\hfill & \hfill sinA+cosB\hfill \\ \hfill sinB\hfill & \hfill cosA\hfill & \hfill sinB+cosB\hfill \\ \hfill sinC\hfill & \hfill cosA\hfill & \hfill sinC+cosB\hfill \end{array}\right|\\ Splittingup{C}_3\\ =\left|\begin{array}{ccc}\hfill sinA\hfill & \hfill cosA\hfill & \hfill sinA\hfill \\ \hfill sinB\hfill & \hfill cosA\hfill & \hfill sinB\hfill \\ \hfill sinC\hfill & \hfill cosA\hfill & \hfill sinC\hfill \end{array}\right|+\left|\begin{array}{ccc}\hfill sinA\hfill & \hfill cosA\hfill & \hfill cosB\hfill \\ \hfill sinB\hfill & \hfill cosA\hfill & \hfill cosB\hfill \\ \hfill sinC\hfill & \hfill cosA\hfill & \hfill cosB\hfill \end{array}\right|\\ =0+\left|\begin{array}{ccc}\hfill sinA\hfill & \hfill cosA\hfill & \hfill cosB\hfill \\ \hfill sinB\hfill & \hfill cosA\hfill & \hfill cosB\hfill \\ \hfill sinC\hfill & \hfill cosA\hfill & \hfill cosB\hfill \end{array}\right|\left[?C_{1}and{C}_{3}areidentical\right]\\ =cosAcosB\left|\begin{array}{ccc}\hfill sinA\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill sinB\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill sinC\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right|\left[TakingcosAandcosBcommonfrom{C}_{2}and{C}_{3}respectively\right]\\ =cosAcosB\left(0\right)\left[?C_{2}and{C}_{3}areidentical\right]\\ =0\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}False.\\ Since?adjA?=?A{?}^{n-1}wherenistheorderofthesquarematrix.\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ Let\Delta =\left|\begin{array}{ccc}\hfill x+1\hfill & \hfill x+2\hfill & \hfill x+a\hfill \\ \hfill x+2\hfill & \hfill x+3\hfill & \hfill x+b\hfill \\ \hfill x+3\hfill & \hfill x+4\hfill & \hfill x+c\hfill \end{array}\right|\\ R_2\to 2R_2-\left(R_1+R_3\right)\\ =\left|\begin{array}{ccc}\hfill x+1\hfill & \hfill x+2\hfill & \hfill x+a\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2b-\left(a+c\right)\hfill \\ \hfill x+3\hfill & \hfill x+4\hfill & \hfill x+c\hfill \end{array}\right|\\ a,b,careinA.P.\\ \therefore b-a=c-b\Rightarrow 2b=a+c\\ =\left|\begin{array}{ccc}\hfill x+1\hfill & \hfill x+2\hfill & \hfill x+a\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill x+3\hfill & \hfill x+4\hfill & \hfill x+c\hfill \end{array}\right|=0\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ \text{\hspace{0.17em}Since}\left|A\right|=12\\ IfAisasquarematrixofordern\\ then\left|AdjA\right|={\left|A\right|}^{n-1}\\ \therefore \left|AdjA\right|={\left|A\right|}^{3-1}={\left|A\right|}^2={\left(12\right)}^2=144\left[n=3\right]\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ \left|3AB\right|=3^3\left|AB\right|=27\left|A\right|\left|B\right|=27*5*3\left[?\left|KA\right|=K^n\left|A\right|\right]\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}False.\\ Since\text{?}\text{?}\left|A^{?1}\right|={\left|A\right|}^{?1}for\text{?}\text{?}a\text{?}\text{?}non?singular\text{?}\text{?}matrix.\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Ifanon-singularsquarematix,thenforanynon-zeroscalar\text{'}a\text{'},aAisinvertible.\\ \therefore \left(aA\right).\left(\frac{1}{a}{A}^{-1}\right)=a.\frac{1}{a}.A.A^{-1}=I\\ So,\left(aA\right)isinverseof\left(\frac{1}{a}{A}^{-1}\right)\\ \Rightarrow {\left(aA\right)}^{-1}=\frac{1}{a}{A}^{-1}istrue.\end{array}$