Question 1

Suppose that
\[
\left|\begin{array}{lll}
a & b & c \\
d & e & f \\
g & h & i
\end{array}\right|=5
\]
then,
\[
\begin{array}{l}
\left|\begin{array}{ccc}
2 a+d & 2 b+e & 2 c+f \\
d & e & f \\
g & h & i
\end{array}\right|=\ldots \text { [Select] } \\
\left|\begin{array}{ccc}
d & e & f \\
a+2 g & b+2 h & c+2 i \\
g & h & i
\end{array}\right|=\ldots \mid[\text { Select] } \\
\left|\begin{array}{ccc}
d & e & f \\
2 g & 2 h & 2 i \\
a & b & c
\end{array}\right|=\ldots
\end{array}
\]

Step 1 ：The determinant of a matrix is a special number that can be calculated from a matrix. The determinant helps us find the inverse of a matrix, tells us things about the matrix that are useful in systems of linear equations, calculus and more.

Step 2 ：The determinant of a 3x3 matrix can be calculated by the rule of Sarrus. However, the determinant has some properties that can simplify the calculation:

Step 3 ：1. The determinant of a matrix remains the same when its rows are switched with columns. In other words, \(\text{det}(A) = \text{det}(A^T)\).

Step 4 ：2. If we multiply a row by a scalar, the determinant is multiplied by this scalar.

Step 5 ：3. If a matrix has two equal rows or columns, its determinant is zero.

Step 6 ：Let's use these properties to calculate the determinants of the given matrices.

Step 7 ：Given that the determinant of the original matrix is 5, we can calculate the determinants of the other matrices as follows:

Step 8 ：For the first matrix, we have multiplied the first row by 2 and added the second row. Therefore, the determinant of the first matrix is \(2 \times 5 = \boxed{10}\).

Step 9 ：For the second matrix, we have added twice the third row to the second row. Therefore, the determinant of the second matrix is \(2 \times 5 = \boxed{10}\).

Step 10 ：For the third matrix, we have swapped the first and third rows. Therefore, the determinant of the third matrix is the same as the original matrix, which is \(\boxed{5}\).