https://file.aiquickdraw.com/mathbot/file/1725678343930uqvy7lrl.jpg

The matrix shown is:

\[
A = \begin{bmatrix}
3 & 9 & 5 \\
1 & 3 & 1 \\
5 & 5 & 2
\end{bmatrix}
\]

You are asked to prove that \( A \cdot I_3 = I_3 \cdot A = A \), where \( I_3 \) is the 3x3 identity matrix.

### Solution Outline:

1. **Identity Matrix**: The 3x3 identity matrix \( I_3 \) is:
   \[
   I_3 = \begin{bmatrix}
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1
   \end{bmatrix}
   \]

2. **Matrix Multiplication**:
   To prove that multiplying matrix \( A \) with the identity matrix \( I_3 \) from both sides results in \( A \), we will compute both \( A \cdot I_3 \) and \( I_3 \cdot A \).

3. **Left Multiplication**: \( A \cdot I_3 \):
   \[
   A \cdot I_3 = \begin{bmatrix}
   3 & 9 & 5 \\
   1 & 3 & 1 \\
   5 & 5 & 2
   \end{bmatrix}
   \cdot
   \begin{bmatrix}
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1
   \end{bmatrix}
   \]
   Using matrix multiplication rules, this results in \( A \).

4. **Right Multiplication**: \( I_3 \cdot A \):
   Similarly,
   \[
   I_3 \cdot A = \begin{bmatrix}
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1
   \end{bmatrix}
   \cdot
   \begin{bmatrix}
   3 & 9 & 5 \\
   1 & 3 & 1 \\
   5 & 5 & 2
   \end{bmatrix}
   \]
   Again, this results in \( A \).

Thus, both \( A \cdot I_3 = A \) and \( I_3 \cdot A = A \), proving that \( A \cdot I_3 = I_3 \cdot A = A \).

Would you like more details or clarification on any step? Here are some related questions:

1. What are the properties of the identity matrix in matrix multiplication?
2. How does matrix multiplication work in general?
3. Can the identity matrix be applied to non-square matrices?
4. What are other special matrices like the identity matrix?
5. How would the proof change if we used a non-identity matrix?

**Tip**: When multiplying any matrix by the identity matrix of the same dimension, the result is always the original matrix.

Show that A⋅I₃ = I₃⋅A = A, where A = [[3, 9, 5], [1, 3, 1], [5, 5, 2]].

Learn how to prove that multiplying a 3x3 matrix A with the identity matrix I₃ results in A from both the left and right. This solution involves matrix multiplication and explores properties of identity matrices. Suitable for students in advanced high school or undergraduate courses in linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation