Question

handwritten step-by-step work legibly and circle your answer. need to show work to receive credit.

1. find the determinant by hand and determine the nature of the solution. please show the algebraic work for points. (referenced from section 4.4/ 2 points)

\\(\

$$\begin{bmatrix}-3 & 2 & -3\\\\0 & -1 & -1\\\\3 & 0 & -3\\end{bmatrix}$$

\\)

Explanation:

Step1: Apply 3x3 determinant formula

For matrix

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

, determinant is $a(ei-fh)-b(di-fg)+c(dh-eg)$
$$\det = (-3)[(-1)(-3)-(-1)(0)] - 2[(0)(-3)-(-1)(3)] + (-3)[(0)(0)-(-1)(3)]$$

Step2: Calculate each bracket term

Simplify the inner products
$$\det = (-3)(3-0) - 2(0+3) + (-3)(0+3)$$

Step3: Compute individual products

Multiply coefficients with bracket results
$$\det = (-3)(3) - 2(3) + (-3)(3)$$

Step4: Sum the terms

Calculate final arithmetic sum
$$\det = -9 - 6 - 9$$

Answer:

$\det = -24$
Since the determinant is not equal to 0, the system of linear equations associated with this matrix has a unique solution.