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:

(-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.