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.)

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.

We start with the determinant of the original matrix given by

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

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:

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

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:

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

Thus, the determinant of the new matrix is:

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

Final Answer