Questions: If the determinant of the matrix a b c d e f g h i is 8, find the determinant of the matrix a b c 3d+g 3e+h 3f+i g h i The determinant of the second matrix is equal to what? (Simplify your answer.)

If the determinant of the matrix

a b c
d e f
g h i

is 8, find the determinant of the matrix

a b c
3d+g 3e+h 3f+i
g h i

The determinant of the second matrix is equal to what? (Simplify your answer.)

Transcript text: If $\left|\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right|=8$, find $\left|\begin{array}{rrr}a & b & c \\ 3 d+g & 3 e+h & 3 f+i \\ g & h & i\end{array}\right|$. $\left|\begin{array}{rrr}a & b & c \\ 3 d+g & 3 e+h & 3 f+i \\ g & h & i\end{array}\right|=\square$ (Simplify your answer.)

Solution

Solution Steps

To solve this problem, we can use properties of determinants. Specifically, we can use the linearity property of determinants, which states that if a row (or column) of a determinant is a linear combination of other rows (or columns), the determinant can be expressed as a sum of determinants. Here, the second row of the new matrix is a linear combination of the original second and third rows. We can express the determinant of the new matrix as a sum of two determinants: one with the second row as $3(d, e, f)$ and the other with the second row as $(g, h, i)$. The first determinant is simply 3 times the original determinant, and the second determinant is zero because it has two identical rows.

Step 1: Understanding the Determinant

We start with the determinant of the original matrix given by

\[ \left|\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right| = 8. \]

Step 2: Applying the Linear Combination Property

We need to find the determinant of the new matrix

\[ \left|\begin{array}{rrr}a & b & c \\ 3d + g & 3e + h & 3f + i \\ g & h & i\end{array}\right|. \]

Using the linearity property of determinants, we can express this determinant as:

\[ \left|\begin{array}{rrr}a & b & c \\ 3d & 3e & 3f \\ g & h & i\end{array}\right| + \left|\begin{array}{rrr}a & b & c \\ g & h & i \\ g & h & i\end{array}\right|. \]

Step 3: Evaluating the Determinants

The first determinant can be simplified as follows:

\[ \left|\begin{array}{rrr}a & b & c \\ 3d & 3e & 3f \\ g & h & i\end{array}\right| = 3 \left|\begin{array}{rrr}a & b & c \\ d & e & f \\ g & h & i\end{array}\right| = 3 \times 8 = 24. \]

The second determinant is zero because it has two identical rows:

\[ \left|\begin{array}{rrr}a & b & c \\ g & h & i \\ g & h & i\end{array}\right| = 0. \]

Step 4: Combining the Results

Thus, the determinant of the new matrix is:

\[ \left|\begin{array}{rrr}a & b & c \\ 3d + g & 3e + h & 3f + i \\ g & h & i\end{array}\right| = 24 + 0 = 24. \]