Exemplar Solution

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

$\begin{array}{l}{A}^{-1}doesnotexistifweapplyoneormorerowoperationswhilefinding{A}^{-1}byelementaryrow\\ operations,obtainallzeroesinoneormorerows.\end{array}$

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

$\begin{array}{l}\left(i\right)Giventhat{A}^{\text{'}}=Aand{B}^{\text{'}}=B\\ LetP=AB\\ P^{\text{'}}={\left(AB\right)}^{\text{'}}\\ =B^{\text{'}}{A}^{\text{'}}\\ =BA\left[?A^{\text{'}}=Aand{B}^{\text{'}}=B\right]\\ =P\\ Hence,ABissymmetricifandonlyifAB=BA.\end{array}$

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

$\begin{array}{l}IfAisasymmetricmatrix\\ \therefore A^{\text{'}}=-A\\ LetP=B^{\text{'}}AB\\ P^{\text{'}}={\left(B^{\text{'}}AB\right)}^{\text{'}}=B^{\text{'}}{A}^{\text{'}}{\left(B^{\text{'}}\right)}^{\text{'}}\left[?{\left(AB\right)}^{\text{'}}=B^{\text{'}}{A}^{\text{'}}\right]\\ =B^{\text{'}}AB\left[?A^{\text{'}}=Aand{\left(B^{\text{'}}\right)}^{\text{'}}=B\right]\\ P^{\text{'}}=P\\ So,Pisasymmetricmatrix.\\ Hence,B^{\text{'}}ABisasymmetricmatrix.\end{array}$

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

$\begin{array}{l}\left(i\right)LetP=\left(AB-BA\right)\\ P^{\text{'}}={\left(AB-BA\right)}^{\text{'}}\\ ={\left(AB\right)}^{\text{'}}-{\left(BA\right)}^{\text{'}}\\ =B^{\text{'}}{A}^{\text{'}}-A^{\text{'}}{B}^{\text{'}}\left[?{\left(AB\right)}^{\text{'}}=B^{\text{'}}{A}^{\text{'}}\right]\\ =BA-AB\left[?A^{\text{'}}=Aand{B}^{\text{'}}=B\right]\\ =-\left(AB-BA\right)=-P\\ Hence,\left(AB-BA\right)isaskewsymmetricmatrix.\\ \left(ii\right)LetQ=\left(BA-2AB\right)\\ Q^{\text{'}}={\left(BA-2AB\right)}^{\text{'}}\\ ={\left(BA\right)}^{\text{'}}-{\left(2AB\right)}^{\text{'}}\\ =A^{\text{'}}{B}^{\text{'}}-2{\left(AB\right)}^{\text{'}}\left[?{\left(kA\right)}^{\text{'}}=k{A}^{\text{'}}\right]\\ =A^{\text{'}}{B}^{\text{'}}-2B^{\text{'}}{A}^{\text{'}}\\ \text{\&}\end{array}$

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

$\begin{array}{l}IfAisaskewsymmetricmatrix\\ \therefore A^{\text{'}}=-A\\ {\left(kA\right)}^{\text{'}}=k{A}^{\text{'}}=k\left(-A\right)=-kA\\ Hence,kAisaskewsymmetricmatrix.\end{array}$

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

$\begin{array}{l}\left(i\right){\left(AB\right)}^{\text{'}}=B^{\text{'}}{A}^{\text{'}}\\ \left(ii\right){\left(kA\right)}^{\text{'}}=k.A^{\text{'}}\\ \left(iii\right){\left[k\left(A-B\right)\right]}^{\text{'}}=k{\left(A-B\right)}^{\text{'}}=k\left(A^{\text{'}}-B^{\text{'}}\right)\end{array}$

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

$\begin{array}{l}IfAisaskewsymmetricmatrix,\\ \therefore A^{\text{'}}=-A\\ {\left(A^2\right)}^{\text{'}}={\left(A^{\text{'}}\right)}^2={\left(-A\right)}^2=A^2\\ Hence,A^{2}isasymmetricmatrix.\end{array}$

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

$\begin{array}{l}LetAbeasymmetricmatrix\\ \therefore A^{\text{'}}=A\\ {\left(A^3\right)}^{\text{'}}={\left(A^{\text{'}}\right)}^3=A^3\left[?{\left(A^{\text{'}}\right)}^k={\left(A^k\right)}^{\text{'}}\right]\\ Hence,ifAisasymmetricmatrix,then{A}^{3}isasymmetricmatrix.\end{array}$

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

$\begin{array}{l}Matrixmultiplicationisdistributiveoveraddition.LetA,B,andCbeanymatrices.\\ So,\left(i\right)A\left(B+C\right)=AB+AC\\ \left(ii\right)\left(A+B\right)C=AC+BC\end{array}$

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

$Amatrixwhichisnotasquarematrixiscalledarectangularmatrix.$