https://file.aiquickdraw.com/mathbot/file/1731003361162vtsrch68.jpg

To show that the matrix \( A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \) is a linear combination of the matrices \( B = \begin{bmatrix} 0 & -1 \\ -1 & 1 \end{bmatrix} \), \( C = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \), and \( D = \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix} \), we need to express \( A \) as:
\[
A = xB + yC + zD
\]
where \( x \), \( y \), and \( z \) are scalars (real numbers) that we need to determine.

This translates to the equation:
\[
\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} = x \begin{bmatrix} 0 & -1 \\ -1 & 1 \end{bmatrix} + y \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} + z \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}
\]

Expanding the right side, we have:
\[
x \begin{bmatrix} 0 & -1 \\ -1 & 1 \end{bmatrix} + y \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} + z \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} y & -x + y + 2z \\ -x + y + 2z & x + y \end{bmatrix}
\]

By comparing corresponding elements in the matrices, we get the following system of equations:
1. \( y = 1 \)
2. \( -x + y + 2z = 2 \)
3. \( x + y = 3 \)

Substituting \( y = 1 \) into equations 2 and 3:

1. \( y = 1 \)
2. \( -x + 1 + 2z = 2 \Rightarrow -x + 2z = 1 \)
3. \( x + 1 = 3 \Rightarrow x = 2 \)

Now substituting \( x = 2 \) into equation 2:

\[
-2 + 2z = 1 \Rightarrow 2z = 3 \Rightarrow z = \frac{3}{2}
\]

So, the solution is:
\[
x = 2, \quad y = 1, \quad z = \frac{3}{2}
\]

Thus, we can express \( A \) as:
\[
A = 2B + 1C + \frac{3}{2}D
\]

This confirms that \( A \) is indeed a linear combination of \( B \), \( C \), and \( D \) with coefficients \( x = 2 \), \( y = 1 \), and \( z = \frac{3}{2} \).

Would you like further details on any specific step, or do you have any other questions?

### Related Questions
1. How can you determine if a set of matrices forms a basis for a vector space?
2. What is the process of finding the rank of a set of matrices?
3. How do you calculate the determinant of a matrix formed by linear combinations?
4. Can every matrix in \( \mathbb{R}^{2 \times 2} \) be written as a combination of matrices \( B \), \( C \), and \( D \)?
5. What are the applications of linear combinations in matrix spaces?

### Tip
When solving systems of linear equations, substituting known values early can simplify the remaining equations significantly.

No espaço vetorial das matrizes reais quadradas de ordem 2, \( \mathbb{R}^{2 \times 2} \), mostre que \( A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \) é combinação linear de \( B = \begin{bmatrix} 0 & -1 \\ -1 & 1 \end{bmatrix} \), \( C = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \), e \( D = \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix} \).

Learn how to express the matrix \( A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \) as a linear combination of matrices \( B = \begin{bmatrix} 0 & -1 \\ -1 & 1 \end{bmatrix} \), \( C = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \), and \( D = \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix} \) in the vector space of \( \mathbb{R}^{2 \times 2} \). This example covers matrix algebra, linear combinations, and vector space 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