(D) Statement $(a)$ is a fundamental property of determinants: if two rows or columns are identical,the determinant is $0$.Statement $(b)$ refers to t

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(D) Statement $(a)$ is a fundamental property of determinants: if two rows or columns are identical,the determinant is $0$.Statement $(b)$ refers to the property that the determinant of a matrix $A$ is equal to the determinant of its transpose $A^T$,i.e.,$|A| = |A^T|$. Thus,interchanging rows and columns does not change the value.Statement $(c)$ is a property stating that swapping any two rows or columns multiplies the determinant by $-1$,thus changing its sign.Therefore,all statements $(a)$,$(b)$,and $(c)$ are correct.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Properties of determinants

Easy

Consider the following statements :
$(a)$ If any two rows or columns of a determinant are identical,then the value of the determinant is zero.
$(b)$ If the corresponding rows and columns of a determinant are interchanged,then the value of the determinant does not change.
$(c)$ If any two rows (or columns) of a determinant are interchanged,then the value of the determinant changes in sign.
Which of these are correct?

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A
$(a)$ and $(b)$
B
$(b)$ and $(c)$
C
$(a)$ and $(c)$
D
$(a)$,$(b)$ and $(c)$

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3 and 4 .Determinants and Matrices

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Properties of determinants

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Let the determinant of a $3 \times 3$ matrix $A$ be $6$. If $B$ is a matrix defined by $B = 5A^2$,then the determinant of $B$ is:

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If $a - 2b + c = 1$,then the value of $\left| \begin{array}{ccc} x + 1 & x + 2 & x + a \\ x + 2 & x + 3 & x + b \\ x + 3 & x + 4 & x + c \end{array} \right|$ is

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If $\omega$ is one of the imaginary cube roots of unity,then the value of the determinant $\left| \begin{array}{ccc} 1 & \omega^3 & \omega^2 \\ \omega^3 & 1 & \omega \\ \omega^2 & \omega & 1 \end{array} \right|$ is:

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Let the determinant of a $3 \times 3$ matrix $A$ be $6$. If $B$ is a matrix defined by $B = 5A^2$,then the determinant of $B$ is:

$\left| {\begin{array}{*{20}{c}} 1 & 5 & \pi \\ {{\log }_e}e & 5 & {\sqrt 5 } \\ {{\log }_{10}}10 & 5 & e \end{array}} \right| = $

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