Find determinant of the given matrix $$ C $$ $$ C=\left(\begin{array}{ccc}-5 & 4 & -5 \ 2 & -6 & 4 \ 6 & 2 & 4\end{array}\right) $$

Question

UpStudy AI · Accepted Answer

To find the determinant of the matrix $$ C $$:

$$
C = \begin{pmatrix}
-5 & 4 & -5 \
2 & -6 & 4 \
6 & 2 & 4
\end{pmatrix}
$$

We can use the **cofactor expansion method** for a $$ 3 \times 3 $$ matrix. The determinant of a matrix

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

is calculated as:

$$
\text{det}(C) = a(ei - fh) - b(di - fg) + c(dh - eg)
$$

**Applying the values from matrix $$ C $$:**

1. **First Term ($$ a(ei - fh) $$):**
   $$
   -5 \times [(-6)(4) - (4)(2)] = -5 \times (-24 - 8) = -5 \times (-32) = 160
   $$

2. **Second Term ($$ -b(di - fg) $$):**
   $$
   -4 \times [(2)(4) - (4)(6)] = -4 \times (8 - 24) = -4 \times (-16) = 64
   $$

3. **Third Term ($$ c(dh - eg) $$):**
   $$
   -5 \times [(2)(2) - (-6)(6)] = -5 \times (4 + 36) = -5 \times 40 = -200
   $$

**Adding all the terms together:**

$$
160 + 64 - 200 = 24
$$

**Conclusion:**
The determinant of matrix $$ C $$ is **24**.

$$\boxed{24}$$
The determinant of matrix $$ C $$ is 24.

Find determinant of the given matrix \( C \) \[ C=\left(\begin{array}{ccc}-5 & 4 & -5 \\ 2 & -6 & 4 \\ 6 & 2 & 4\end{array}\right) \]