If $$ M=\left[\begin{array}{lll}a & b & c \ d & e & f \ g & h & i\end{array} ight] $$ and $$ \operatorname{det}(M)=4 $$ $$ \operatorname{Det}\left(\left[\begin{array}{lll}g & h & i \ d & e & f \ a & b & c\end{array} ight] ight)=\square $$

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If $$ M=\left[\begin{array}{lll}a & b & c \ d & e & f \ g & h & i\end{array}ight] $$ and $$ \operatorname{det}(M)=4 $$ $$ \operatorname{Det}\left(\left[\begin{array}{lll}g & h & i \ d & e & f \ a & b & c\end{array}ight]ight)=\square $$

UpStudy AI · Accepted Answer

To find the determinant of the matrix

$$
\left[\begin{array}{lll}g & h & i \ d & e & f \ a & b & c\end{array}\right],
$$

we can use the property of determinants that states that swapping two rows of a matrix changes the sign of the determinant.

Given that

$$
M=\left[\begin{array}{lll}a & b & c \ d & e & f \ g & h & i\end{array}\right]
$$

has a determinant of $$ \operatorname{det}(M) = 4 $$, we can find the determinant of the new matrix by performing the following row swaps:

1. Swap the first row with the third row.
2. Swap the second row with the first row.

Each swap will change the sign of the determinant.

Let's calculate the determinant step by step:

1. Swap the first and third rows: 
   $$
   \operatorname{det}\left(\left[\begin{array}{lll}g & h & i \ d & e & f \ a & b & c\end{array}\right]\right) = -\operatorname{det}(M) = -4.
   $$

2. Now, swap the second and first rows again:
   $$
   \operatorname{det}\left(\left[\begin{array}{lll}g & h & i \ d & e & f \ a & b & c\end{array}\right]\right) = -(-4) = 4.
   $$

Thus, the determinant of the new matrix is $$ 4 $$.

So, the answer is:

$$
\operatorname{Det}\left(\left[\begin{array}{lll}g & h & i \ d & e & f \ a & b & c\end{array}\right]\right) = 4.
$$
The determinant of the new matrix is 4.

If and

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