Question 9

What is the determinant of the matrix $\left[\begin{array}{ccc}1 & 0 & -3 \\ -4 & 7 & 5 \\ 2 & -8 & 6\end{array}\right]$ ?

Question 10

Step 1 ：We are given the matrix \(A = \left[\begin{array}{ccc}1 & 0 & -3 \\ -4 & 7 & 5 \\ 2 & -8 & 6\end{array}\right]\)

Step 2 ：The determinant of a 3x3 matrix can be calculated using the formula: \(\text{det}(A) = a(ei−fh)−b(di−fg)+c(dh−eg)\)

Step 3 ：Substituting the given values into the formula, we get: \(\text{det}(A) = 1(7*6 - 5*(-8)) - 0*(-4*6 - 5*2) - 3*(-4*(-8) - 7*2)\)

Step 4 ：Solving the above expression, we get: \(\text{det}(A) = 1(42 + 40) - 0 - 3(32 - 14)\)

Step 5 ：Simplifying further, we get: \(\text{det}(A) = 1*82 - 3*18\)

Step 6 ：Finally, we get: \(\text{det}(A) = 82 - 54 = 28\)

Step 7 ：Final Answer: The determinant of the matrix is \(\boxed{28}\)