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If $A$ and $B$ are square matrices of size $n\times n$, then which of the following statements is not true?

  1. $\det(AB)=\det(A) \det(B)$
  2. $\det(kA)=k^n \det(A)$
  3. $\det(A+B)=\det(A)+\det(B)$
  4. $\det(A^T)=1/\det(A^{-1})$
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statement " det(A+B)=det(A)+det(B) " is not correct,

consider A = I; det(A) = 1; //I is Identity matrix of order 2X2

B = -A = -I; det(B) = 1; //det(X) returns determinant of X

det(A+B) = 0;

while, det(A) + det(B) = 2;

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A counter example.

Let $A = \begin{bmatrix} 1 &0 \\ 0 &0 \end{bmatrix}$ and $B = \begin{bmatrix} 0 &0 \\ 0 & 1 \end{bmatrix}$

Then, $det\bigl(\begin{smallmatrix} \begin{bmatrix} 1 &0 \\ 0 &0 \end{bmatrix} + &\begin{bmatrix} 0 &0 \\ 0 & 1 \end{bmatrix} \end{smallmatrix}\bigr)= det\begin{pmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{pmatrix} = 1$

But, $det\begin{pmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \end{pmatrix} + det\begin{pmatrix} \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \end{pmatrix} = 0+0 = 0$

Therefore, $det\begin{pmatrix} A +B \end{pmatrix} \neq det\begin{pmatrix} A \end{pmatrix} + det\begin{pmatrix} B \end{pmatrix}$
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