Find the determinant of the matrix  M=[1,0,0,2],[3,0,-1,0],[0,-3,0,-1],[0,2,-2,0]   det(M)=

We are tasked with finding the determinant of the matrix \( M \):

\[
M = \begin{pmatrix}
1 & 0 & 0 & 2 \\
3 & 0 & -1 & 0 \\
0 & -3 & 0 & -1 \\
0 & 2 & -2 & 0
\end{pmatrix}
\]

### Step-by-step calculation:

To find the determinant of this 4x4 matrix, we can use cofactor expansion along the first row (since the first row has simple entries). The determinant of \( M \) is calculated as:

\[
\det(M) = 1 \cdot \det\left(\begin{pmatrix}
0 & -1 & 0 \\
-3 & 0 & -1 \\
2 & -2 & 0
\end{pmatrix}\right) 
- 0 \cdot \det\left(\begin{pmatrix}
3 & -1 & 0 \\
0 & 0 & -1 \\
0 & -2 & 0
\end{pmatrix}\right) 
+ 0 \cdot \det\left(\begin{pmatrix}
3 & 0 & 0 \\
0 & -3 & -1 \\
0 & 2 & -2
\end{pmatrix}\right) 
+ 2 \cdot \det\left(\begin{pmatrix}
3 & 0 & -1 \\
0 & -3 & 0 \\
0 & 2 & -2
\end{pmatrix}\right)
\]

Notice that two of the terms vanish because of the zeros in the second and third positions in the first row. Now, we just need to calculate the two remaining 3x3 determinants.

#### First 3x3 determinant:

\[
\det\left(\begin{pmatrix}
0 & -1 & 0 \\
-3 & 0 & -1 \\
2 & -2 & 0
\end{pmatrix}\right)
\]

We can compute this determinant by cofactor expansion along the first row:

\[
= 0 \cdot \det\begin{pmatrix} 0 & -1 \\ -2 & 0 \end{pmatrix} - (-1) \cdot \det\begin{pmatrix} -3 & -1 \\ 2 & 0 \end{pmatrix} + 0 \cdot \det\begin{pmatrix} -3 & 0 \\ 2 & -2 \end{pmatrix}
\]

\[
= \det\begin{pmatrix} -3 & -1 \\ 2 & 0 \end{pmatrix}
\]

\[
= (-3)(0) - (-1)(2) = 2
\]

#### Second 3x3 determinant:

\[
\det\left(\begin{pmatrix}
3 & 0 & -1 \\
0 & -3 & 0 \\
0 & 2 & -2
\end{pmatrix}\right)
\]

We expand this along the first row:

\[
= 3 \cdot \det\begin{pmatrix} -3 & 0 \\ 2 & -2 \end{pmatrix} - 0 \cdot \det\begin{pmatrix} 0 & 0 \\ 0 & -2 \end{pmatrix} + (-1) \cdot \det\begin{pmatrix} 0 & -3 \\ 0 & 2 \end{pmatrix}
\]

\[
= 3 \cdot \left((-3)(-2) - (0)(2)\right) + (-1) \cdot \left(0 \cdot 2 - 0 \cdot (-3)\right)
\]

\[
= 3 \cdot 6 = 18
\]

#### Putting it all together:

Now we substitute back into the original cofactor expansion:

\[
\det(M) = 1 \cdot 2 + 2 \cdot 18 = 2 + 36 = 38
\]

Thus, the determinant of the matrix \( M \) is:

\[
\boxed{38}
\]

---

Would you like any more details on this? 

Here are some related questions to help deepen your understanding:

1. How would the determinant change if we expanded along a different row or column?
2. How do properties like row swaps or scalar multiplication affect the determinant?
3. What is the geometric interpretation of the determinant of a matrix?
4. How would you compute the determinant of a 3x3 matrix using the Sarrus rule?
5. What are the implications of a matrix having a determinant of 0?

Tip: To make determinant calculations easier, look for rows or columns with many zeros to expand along!

Find the determinant of the matrix M=[1,0,0,2],[3,0,-1,0],[0,-3,0,-1],[0,2,-2,0]

Learn how to find the determinant of a 4x4 matrix using cofactor expansion, including step-by-step calculations for matrix M. This solution is designed for high school students studying linear algebra, covering key concepts like cofactor expansion and matrix determinants.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation