Determinants

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:

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

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

Sol:

$\begin{array}{l}Since{\left(A^K\right)}^{-1}={\left(A^{-1}\right)}^{K}whereK\in N\\ So,{\left(A^3\right)}^{-1}={\left(A^{-1}\right)}^{3}istrue\end{array}$

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

Sol:

$\begin{array}{l}Giventhat \left[\begin{array}{ccc}\hfill {\left(1+x\right)}^{17}\hfill & \hfill {\left(1+x\right)}^{19}\hfill & \hfill {\left(1+x\right)}^{23}\hfill \\ \hfill {\left(1+x\right)}^{23}\hfill & \hfill {\left(1+x\right)}^{29}\hfill & \hfill {\left(1+x\right)}^{34}\hfill \\ \hfill {\left(1+x\right)}^{41}\hfill & \hfill {\left(1+x\right)}^{43}\hfill & \hfill {\left(1+x\right)}^{47}\hfill \end{array}\right]=A+Bx+C{x}^2+\dots \\ Taking{\left(1+x\right)}^{17},{\left(1+x\right)}^{23}and{\left(1+x\right)}^{41}commonfrom{R}_1,R_{2}and{R}_{3}respectively\\ {\left(1+x\right)}^{17}.{\left(1+x\right)}^{23}.{\left(1+x\right)}^{41}\left|\begin{array}{ccc}\hfill 1\hfill & \hfill {\left(1+x\right)}^2\hfill & \hfill {\left(1+x\right)}^6\hfill \\ \hfill 1\hfill & \hfill {\left(1+x\right)}^6\hfill & \hfill {\left(1+x\right)}^{11}\hfill \\ \hfill 1\hfill & \hfill {\left(1+x\right)}^2\hfill & \hfill {\left(1+x\right)}^6\hfill \end{array}\right|\\ \Rightarrow {\left(1+x\right)}^{17}.{\left(1+x\right)}^{23}.{\left(1+x\right)}^{41}.0\left(R_{1}and{R}_{3}areidentical\right)\\ \therefore 0=A+Bx+C{x}^2+\dots \\ Bycomparingtheliketerms,wegetA=0.\end{array}$

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

Sol:

$\begin{array}{l}Let\Delta =\left|\begin{array}{ccc}\hfill 0\hfill & \hfill xyz\hfill & \hfill x-z\hfill \\ \hfill y-x\hfill & \hfill 0\hfill & \hfill y-z\hfill \\ \hfill z-x\hfill & \hfill z-y\hfill & \hfill 0\hfill \end{array}\right|\\ C_1\to C_1-C_3\\ =\left|\begin{array}{ccc}\hfill z-x\hfill & \hfill xyz\hfill & \hfill x-z\hfill \\ \hfill z-x\hfill & \hfill 0\hfill & \hfill y-z\hfill \\ \hfill z-x\hfill & \hfill z-y\hfill & \hfill 0\hfill \end{array}\right|\\ Taking\left(z-x\right)commonfrom{C}_1\\ =\left(z-x\right)\left|\begin{array}{ccc}\hfill 1\hfill & \hfill xyz\hfill & \hfill x-z\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill y-z\hfill \\ \hfill 1\hfill & \hfill z-y\hfill & \hfill 0\hfill \end{array}\right|\\ R_1\to R_1-R_2,R_2\to R_2-R_3\\ =\left(z-x\right)\left|\begin{array}{ccc}\hfill 0\hfill & \hfill xyz\hfill & \hfill x-y\hfill \\ \hfill 0\hfill & \hfill y-z\hfill & \hfill y-z\hfill \\ \hfill 1\hfill & \hfill z-y\hfill & \hfill 0\hfill \end{array}\right|\\ Taking\left(y-z\right)commonfrom{R}_2\\ =\left(z-x\right)\left(y-z\right)\left|\begin{array}{ccc}\hfill 0\hfill & \hfill xyz\hfill & \hfill x-y\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill z-y\hfill & \hfill 0\hfill \end{array}\right|\\ Expandingalong{C}_1\\ =\left(z-x\right)\left(y-z\right)\left[1\left|\begin{array}{cc}\hfill xyz\hfill & \hfill x-y\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right|\right]\\ =\left(z-x\right)\left(y-z\right)\left(xyz-x+y\right)=\left(y-z\right)\left(z-x\right)\left(y-x+xyz\right)\end{array}$

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

Sol:

$\begin{array}{l}Wehave,\left|\begin{array}{ccc}\hfill x\hfill & \hfill 3\hfill & \hfill 7\hfill \\ \hfill 2\hfill & \hfill x\hfill & \hfill 2\hfill \\ \hfill 7\hfill & \hfill 6\hfill & \hfill x\hfill \end{array}\right|=0\\ Expandingalong{R}_1\\ \Rightarrow x\left|\begin{array}{cc}\hfill x\hfill & \hfill 2\hfill \\ \hfill 6\hfill & \hfill x\hfill \end{array}\right|-3\left|\begin{array}{cc}\hfill 2\hfill & \hfill 2\hfill \\ \hfill 7\hfill & \hfill x\hfill \end{array}\right|+7\left|\begin{array}{cc}\hfill 2\hfill & \hfill x\hfill \\ \hfill 7\hfill & \hfill 6\hfill \end{array}\right|=0\\ \Rightarrow x\left(x^2-12\right)-3\left(2x-14\right)+7\left(12-7x\right)=0\\ \Rightarrow x^3-12x-6x+42+84-49x=0\\ \Rightarrow x^3-67x+126=0\dots \left(1\right)\\ Therootsoftheequationmaybethefactorsof126i.e.,2*7*9\\ 9isthegivenrootofthedeterminantputx=2ineq.\left(1\right)\\ {\left(2\right)}^3-67*2+126\Rightarrow 8-134+126=0\\ Hence,x=2istheotherroot.\\ Now,putx=7ineq.\left(1\right)\\ {\left(7\right)}^3-67*7+126\Rightarrow 343-469+126=0\\ Hence,x=7isalsotheotherrootofthedeterminant.\end{array}$

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

Sol:

$\begin{array}{l}Thesumoftheproductsofelementsofanyrowwiththeco-factorsofcorrespondingelements\\ isequaltothevalueofthedeterminantofthegivenmatrix.\\ Let\Delta =\left|\begin{array}{ccc}\hfill a_{11}\hfill & \hfill a_{12}\hfill & \hfill a_{13}\hfill \\ \hfill a_{21}\hfill & \hfill a_{22}\hfill & \hfill a_{23}\hfill \\ \hfill a_{31}\hfill & \hfill a_{32}\hfill & \hfill a_{33}\hfill \end{array}\right|\\ Expandingalong{R}_1\\ a_{11}\left|\begin{array}{cc}\hfill a_{22}\hfill & \hfill a_{23}\hfill \\ \hfill a_{32}\hfill & \hfill a_{33}\hfill \end{array}\right|-a_{12}\left|\begin{array}{cc}\hfill a_{21}\hfill & \hfill a_{23}\hfill \\ \hfill a_{31}\hfill & \hfill a_{33}\hfill \end{array}\right|+a_{13}\left|\begin{array}{cc}\hfill a_{21}\hfill & \hfill a_{22}\hfill \\ \hfill a_{31}\hfill & \hfill a_{32}\hfill \end{array}\right|\\ \Rightarrow a_{11}{M}_{11}+a_{12}{M}_{12}+a_{13}{M}_{13}\left(where{M}_{11},M_{12}and{M}_{13}aretheminorsofthecorrespondingelements\right)\end{array}$

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

Sol:

$\begin{array}{l}Theorderofamatrixis3*3\\ \therefore Totalnumberofelements=3*3=9\\ Hence,thenumberofminorsinthedeterminantis9.\end{array}$

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

Sol: $ForanysquarematrixA,{\left(A^2\right)}^{-1}={\left(A^{-1}\right)}^2.$

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

Sol: