Question-6

Let \(A,B,C\) be three matrices of order \(3\). Comment on the truth value of the following statements:

1. \(\text{det}(ABC)=\text{det}(A)\text{det}(B)\text{det}(C)\)

2. \(\text{det}\left(A^{3}\right)=\left(\text{det}(A)\right)^{3}\)

3. \(\text{det}(A+B+C)=\text{det}(A)+\text{det}(B)+\text{det}(C)\)

4. \(\text{det}\left(AB^{T}\right)=\text{det}(A)\text{det}(B)\)

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(1) We have:

\[ \begin{aligned} \text{det}(ABC) & =\text{det}((AB)C)\\ & =\text{det}(AB)\text{det}(C)\\ & =\text{det}(A)\text{det}(B)\text{det}(C) \end{aligned} \]

(2) Using the previous result and setting \(A=B=C\) shows that \(\text{det}(A^{3})=\text{det}(A)^{3}\).

(3) This is not true. Here is a counter-example:

\[ \begin{aligned} A=B=C= & I\\ \implies A+B+C & =3I\\ \implies\text{det}(A+B+C) & =27\\ \implies\text{det}(A)+\text{det}(B)+\text{det}(C) & =3 \end{aligned} \]

(4) This result is true.

\[ \begin{aligned} \text{det}(AB^{T}) & =\text{det}(A)\text{det}(B^{T})=\text{det}(A)\text{det}(B) \end{aligned} \]