Question

question 8
5 pts
simplify each problem. find the
determinant of each matrix first.
write the final solution in the
answer blank:
\

$$\begin{vmatrix} 3 & 5 & 1 \\\\ -4 & 1 & -1 \\\\ 3 & 2 & -4 \\end{vmatrix}$$

• \

$$\begin{vmatrix} 3 & 5 & -4 \\\\ -1 & 2 & -4 \\\\ -4 & -5 & 4 \\end{vmatrix}$$

• \

$$\begin{vmatrix} -4 & 1 & 1 \\\\ 3 & 5 & -4 \\\\ -4 & 3 & 5 \\end{vmatrix}$$

Explanation:

Step1: Calculate determinant of first matrix

For matrix \(

$$\begin{vmatrix} 3 & 5 & 1 \\ -4 & 1 & -1 \\ 3 & 2 & -4 \end{vmatrix}$$

\), use the formula for 3x3 determinant \(

$$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$$

=a(ei - fh)-b(di - fg)+c(dh - eg) \).

$$
LATEXBLOCK2
$$

Step2: Calculate determinant of second matrix

For matrix \(

$$\begin{vmatrix} 3 & 5 & -4 \\ -1 & 2 & -4 \\ -4 & -5 & 4 \end{vmatrix}$$

\), apply the 3x3 determinant formula:

$$
LATEXBLOCK4
$$

Step3: Calculate determinant of third matrix

For matrix \(

$$\begin{vmatrix} -4 & 1 & 1 \\ 3 & 5 & -4 \\ -4 & 3 & 5 \end{vmatrix}$$

\), use the determinant formula:

$$
LATEXBLOCK6
$$

Step4: Combine the determinants

The expression is \( (-112)-12-(-118) \) (note the last term is subtracted, so it's - ( - 118)).

$$
LATEXBLOCK7
$$

Answer:

\(-6\)