Statement -1 : Determinant of a skew-symmetric matrix of order 3 is zero Statement -2 :

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3 and 4 .Determinants and Matrices

medium

Statement $-1 :$Determinant of a skew-symmetric matrix of order $3$ is zero

Statement $-2 :$ For any matrix $A,$ $\det \left( {{A^T}} \right) = {\rm{det}}\left( A \right)$ and $\det \left( { - A} \right) = - {\rm{det}}\left( A \right)$ Where $\det \left( A \right) = A$. Then :

Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is a correct explanation for Statement $-1$

Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is not a correct explanation for Statement $-1$

Statement $-1$ is false, Statement $-2$ is true

Statement $-1$ is true, Statement $-2$ is false

(AIEEE-2011)

Solution

Statement $- 1:$ Determinant of a skew sysmmetric matrix of odd order is zero Statement $-$ $2: \operatorname{det}\left(A^{T}\right)=\operatorname{det}(A)$ $\operatorname{det}(-A)=(-1)^{n} \operatorname{det}(A)$ where $\mathrm{A}$ is an $\times$ norder matrix

Standard 12

Mathematics

If $A=\left[\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right],$ then show that $|2 A|=4|A|$.

easy

Let $A=\left(\begin{array}{cc}4 & -2 \\ \alpha & \beta\end{array}\right)$ . If $A ^{2}+\gamma A +18 I = O$, then $\operatorname{det}( A )$ is equal to

easy

(JEE MAIN-2022)

If $x + y – z = 0,\,3x – \alpha y – 3z = 0,\,\,x – 3y + z = 0$ has non zero solution, then $\alpha = $

medium

If the system of equations

$ 11 x+y+\lambda z=-5 $

$ 2 x+3 y+5 z=3 $

$ 8 x-19 y-39 z=\mu$

has infinitely many solutions, then $\lambda^4-\mu$ is equal to :

hard

(JEE MAIN-2024)

If ${2^{{a_1}}},{2^{{a_2}}},{2^{{a_3}}},{……2^{{a_n}}}$ are in $G.P.$ then $\left| {\begin{array}{*{20}{c}}
{{a_1}}&{{a_2}}&{{a_3}} \\
{{a_{n + 1}}}&{{a_{n + 2}}}&{{a_{n + 3}}} \\
{{a_{2n + 1}}}&{{a_{2n + 2}}}&{{a_{2n + 3}}}
\end{array}} \right|$ is equal to

normal

Statement $-1 :$Determinant of a skew-symmetric matrix of order $3$ is zero Statement $-2 :$ For any matrix $A,$ $\det \left( {{A^T}} \right) = {\rm{det}}\left( A \right)$ and $\det \left( { - A} \right) = - {\rm{det}}\left( A \right)$ Where $\det \left( A \right) = A$. Then :

Solution

Similar Questions

If $A=\left[\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right],$ then show that $|2 A|=4|A|$.

Let $A=\left(\begin{array}{cc}4 & -2 \\ \alpha & \beta\end{array}\right)$ . If $A ^{2}+\gamma A +18 I = O$, then $\operatorname{det}( A )$ is equal to

If $x + y – z = 0,\,3x – \alpha y – 3z = 0,\,\,x – 3y + z = 0$ has non zero solution, then $\alpha = $

If the system of equations $ 11 x+y+\lambda z=-5 $ $ 2 x+3 y+5 z=3 $ $ 8 x-19 y-39 z=\mu$ has infinitely many solutions, then $\lambda^4-\mu$ is equal to :

If ${2^{{a_1}}},{2^{{a_2}}},{2^{{a_3}}},{……2^{{a_n}}}$ are in $G.P.$ then $\left| {\begin{array}{*{20}{c}} {{a_1}}&{{a_2}}&{{a_3}} \\ {{a_{n + 1}}}&{{a_{n + 2}}}&{{a_{n + 3}}} \\ {{a_{2n + 1}}}&{{a_{2n + 2}}}&{{a_{2n + 3}}} \end{array}} \right|$ is equal to

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Statement $-1 :$Determinant of a skew-symmetric matrix of order $3$ is zero

Statement $-2 :$ For any matrix $A,$ $\det \left( {{A^T}} \right) = {\rm{det}}\left( A \right)$ and $\det \left( { - A} \right) = - {\rm{det}}\left( A \right)$ Where $\det \left( A \right) = A$. Then :

If the system of equations

$ 11 x+y+\lambda z=-5 $

$ 2 x+3 y+5 z=3 $

$ 8 x-19 y-39 z=\mu$

has infinitely many solutions, then $\lambda^4-\mu$ is equal to :

If ${2^{{a_1}}},{2^{{a_2}}},{2^{{a_3}}},{……2^{{a_n}}}$ are in $G.P.$ then $\left| {\begin{array}{*{20}{c}}
{{a_1}}&{{a_2}}&{{a_3}} \\
{{a_{n + 1}}}&{{a_{n + 2}}}&{{a_{n + 3}}} \\
{{a_{2n + 1}}}&{{a_{2n + 2}}}&{{a_{2n + 3}}}
\end{array}} \right|$ is equal to