(A) For a square matrix $A$ of order $n$,the property of the determinant is $det(kA) = k^n det(A)$.Given that $A$ is a square matrix of order $n = 3$,

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(A) For a square matrix $A$ of order $n$,the property of the determinant is $det(kA) = k^n det(A)$.Given that $A$ is a square matrix of order $n = 3$,we have $det(-A) = det((-1)A)$.Using the property $det(kA) = k^n det(A)$ with $k = -1$ and $n = 3$,we get:$det(-A) = (-1)^3 det(A) = -1 \times det(A) = -det(A)$.Therefore,the statement $det(-A) = -det(A)$ is true.

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

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If $A$ is a square matrix of order $3$,then which of the following statements is true? (where $I$ is the identity matrix)

A
$det(-A) = -det(A)$
B
$det(A) = 0$
C
$det(A + I) = 1 + det(A)$
D
$det(2A) = 2det(A)$

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3 and 4 .Determinants and Matrices

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Properties of determinants

If $\left|\begin{array}{ccc}a^{2} & b c & c^{2}+a c \\ a^{2}+a b & b^{2} & c a \\ a b & b^{2}+b c & c^{2}\end{array}\right|=k a^{2} b^{2} c^{2}$,then $k=$

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Prove that $\left|\begin{array}{ccc}a^{2} & b c & a c+c^{2} \\ a^{2}+a b & b^{2} & a c \\ a b & b^{2}+b c & c^{2}\end{array}\right|=4 a^{2} b^{2} c^{2}$

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If $A$ is a square matrix of order $3$,then which of the following statements is true? (where $I$ is the identity matrix)

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Using the property of determinants and without expanding,prove that:
$\left|\begin{array}{lll}2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86\end{array}\right|=0$