https://file.aiquickdraw.com/mathbot/file/1726390846084moq0jluu.jpg

The given matrix is:

\[
\begin{bmatrix}
2 & 1 & 0 \\
0 & 2 & 1 \\
0 & 0 & 2
\end{bmatrix}
\]

To find the sum of minors, we first need to compute the minors for each element of the matrix. The **minor** of an element is the determinant of the matrix obtained by deleting the row and column that contains that element.

Let's go step by step:

1. **Minor of the element at position (1,1)** (value = 2):
   We delete the first row and first column to get the matrix:
   \[
   \begin{bmatrix}
   2 & 1 \\
   0 & 2
   \end{bmatrix}
   \]
   The determinant of this matrix is:
   \[
   \text{det} = (2)(2) - (1)(0) = 4
   \]

2. **Minor of the element at position (1,2)** (value = 1):
   We delete the first row and second column to get the matrix:
   \[
   \begin{bmatrix}
   0 & 1 \\
   0 & 2
   \end{bmatrix}
   \]
   The determinant of this matrix is:
   \[
   \text{det} = (0)(2) - (1)(0) = 0
   \]

3. **Minor of the element at position (1,3)** (value = 0):
   We delete the first row and third column to get the matrix:
   \[
   \begin{bmatrix}
   0 & 2 \\
   0 & 0
   \end{bmatrix}
   \]
   The determinant of this matrix is:
   \[
   \text{det} = (0)(0) - (2)(0) = 0
   \]

4. **Minor of the element at position (2,1)** (value = 0):
   We delete the second row and first column to get the matrix:
   \[
   \begin{bmatrix}
   1 & 0 \\
   0 & 2
   \end{bmatrix}
   \]
   The determinant of this matrix is:
   \[
   \text{det} = (1)(2) - (0)(0) = 2
   \]

5. **Minor of the element at position (2,2)** (value = 2):
   We delete the second row and second column to get the matrix:
   \[
   \begin{bmatrix}
   2 & 0 \\
   0 & 2
   \end{bmatrix}
   \]
   The determinant of this matrix is:
   \[
   \text{det} = (2)(2) - (0)(0) = 4
   \]

6. **Minor of the element at position (2,3)** (value = 1):
   We delete the second row and third column to get the matrix:
   \[
   \begin{bmatrix}
   2 & 1 \\
   0 & 0
   \end{bmatrix}
   \]
   The determinant of this matrix is:
   \[
   \text{det} = (2)(0) - (1)(0) = 0
   \]

7. **Minor of the element at position (3,1)** (value = 0):
   We delete the third row and first column to get the matrix:
   \[
   \begin{bmatrix}
   1 & 0 \\
   2 & 1
   \end{bmatrix}
   \]
   The determinant of this matrix is:
   \[
   \text{det} = (1)(1) - (0)(2) = 1
   \]

8. **Minor of the element at position (3,2)** (value = 0):
   We delete the third row and second column to get the matrix:
   \[
   \begin{bmatrix}
   2 & 0 \\
   0 & 1
   \end{bmatrix}
   \]
   The determinant of this matrix is:
   \[
   \text{det} = (2)(1) - (0)(0) = 2
   \]

9. **Minor of the element at position (3,3)** (value = 2):
   We delete the third row and third column to get the matrix:
   \[
   \begin{bmatrix}
   2 & 1 \\
   0 & 2
   \end{bmatrix}
   \]
   The determinant of this matrix is:
   \[
   \text{det} = (2)(2) - (1)(0) = 4
   \]

### Sum of all minors:

\[
4 + 0 + 0 + 2 + 4 + 0 + 1 + 2 + 4 = 17
\]

So, the sum of all minors is **17**.

Would you like further explanations or have any questions?

Here are some related questions to explore:

1. How do you calculate the determinant of a matrix?
2. What is the difference between a minor and a cofactor?
3. How do you find the adjugate of a matrix?
4. What is the role of minors in computing the inverse of a matrix?
5. How does the determinant affect the invertibility of a matrix?

**Tip:** Minors are useful when finding the determinant of larger matrices by expanding along rows or columns.

Find the sum of minors of the given matrix.

Learn how to calculate the sum of minors for a 3x3 matrix. This example explains step-by-step how to compute the minors of each element and find their sum. Ideal for advanced high school or undergraduate students studying matrix theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation