
If A=[(1,0,3),(-2,4,5),(2,0,1)] then find A^2−2A+5I.

Given the matrix \( A = \begin{pmatrix} 1 & 0 & 3 \\ -2 & 4 & 5 \\ 2 & 0 & 1 \end{pmatrix} \), we are asked to find the expression \( A^2 - 2A + 5I \), where \( I \) is the identity matrix.

### Step 1: Calculate \( A^2 \)
To calculate \( A^2 \), we need to multiply the matrix \( A \) by itself:

\[
A^2 = A \times A = \begin{pmatrix} 1 & 0 & 3 \\ -2 & 4 & 5 \\ 2 & 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 0 & 3 \\ -2 & 4 & 5 \\ 2 & 0 & 1 \end{pmatrix}
\]

Perform the matrix multiplication:

\[
A^2 = \begin{pmatrix} (1)(1) + (0)(-2) + (3)(2) & (1)(0) + (0)(4) + (3)(0) & (1)(3) + (0)(5) + (3)(1) \\
(-2)(1) + (4)(-2) + (5)(2) & (-2)(0) + (4)(4) + (5)(0) & (-2)(3) + (4)(5) + (5)(1) \\
(2)(1) + (0)(-2) + (1)(2) & (2)(0) + (0)(4) + (1)(0) & (2)(3) + (0)(5) + (1)(1) \end{pmatrix}
\]

Simplifying each element:

\[
A^2 = \begin{pmatrix} 1 + 0 + 6 & 0 + 0 + 0 & 3 + 0 + 3 \\
-2 - 8 + 10 & 0 + 16 + 0 & -6 + 20 + 5 \\
2 + 0 + 2 & 0 + 0 + 0 & 6 + 0 + 1 \end{pmatrix} = \begin{pmatrix} 7 & 0 & 6 \\ 0 & 16 & 19 \\ 4 & 0 & 7 \end{pmatrix}
\]

### Step 2: Calculate \( -2A \)
Next, calculate \( -2A \):

\[
-2A = -2 \times \begin{pmatrix} 1 & 0 & 3 \\ -2 & 4 & 5 \\ 2 & 0 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 0 & -6 \\ 4 & -8 & -10 \\ -4 & 0 & -2 \end{pmatrix}
\]

### Step 3: Calculate \( 5I \)
The identity matrix \( I \) for a 3x3 matrix is:

\[
I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\]

So, \( 5I \) is:

\[
5I = 5 \times \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix}
\]

### Step 4: Combine the results
Now, combine the results to get \( A^2 - 2A + 5I \):

\[
A^2 - 2A + 5I = \begin{pmatrix} 7 & 0 & 6 \\ 0 & 16 & 19 \\ 4 & 0 & 7 \end{pmatrix} - \begin{pmatrix} -2 & 0 & -6 \\ 4 & -8 & -10 \\ -4 & 0 & -2 \end{pmatrix} + \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix}
\]

First, add \( A^2 \) and \( -2A \):

\[
\begin{pmatrix} 7 & 0 & 6 \\ 0 & 16 & 19 \\ 4 & 0 & 7 \end{pmatrix} + \begin{pmatrix} -2 & 0 & -6 \\ 4 & -8 & -10 \\ -4 & 0 & -2 \end{pmatrix} = \begin{pmatrix} 5 & 0 & 0 \\ 4 & 8 & 9 \\ 0 & 0 & 5 \end{pmatrix}
\]

Now add \( 5I \):

\[
\begin{pmatrix} 5 & 0 & 0 \\ 4 & 8 & 9 \\ 0 & 0 & 5 \end{pmatrix} + \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix} = \begin{pmatrix} 10 & 0 & 0 \\ 4 & 13 & 9 \\ 0 & 0 & 10 \end{pmatrix}
\]

### Final Answer
\[
A^2 - 2A + 5I = \begin{pmatrix} 10 & 0 & 0 \\ 4 & 13 & 9 \\ 0 & 0 & 10 \end{pmatrix}
\]

Would you like any details or have any questions?

Here are five related questions to expand on this topic:
1. How do you calculate the determinant of the resulting matrix \( A^2 - 2A + 5I \)?
2. What is the trace of the matrix \( A^2 - 2A + 5I \)?
3. How can you find the eigenvalues of the matrix \( A^2 - 2A + 5I \)?
4. What is the inverse of the matrix \( A^2 - 2A + 5I \) if it exists?
5. Can the result \( A^2 - 2A + 5I \) be used to determine if \( A \) is diagonalizable?

**Tip:** Always double-check matrix multiplication by computing the individual elements separately to avoid errors.

Learn how to calculate the expression A^2 - 2A + 5I for the given matrix A = [(1, 0, 3), (-2, 4, 5), (2, 0, 1)] in matrix algebra. Follow a detailed step-by-step solution including matrix multiplication and identity matrix operations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation