If A = [1 2; 3 1] show that A² - 2A - 5I = 0 where I is the 2x2 unit matrix.

Steps to Solve

1. Calculate A²

First, compute the square of matrix $A$.

Given: $$ A = \begin{bmatrix} 1 & 2 \ 3 & 1 \end{bmatrix} $$

Calculating $A^2$: $$ A^2 = A \times A = \begin{bmatrix} 1 & 2 \ 3 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \ 3 & 1 \end{bmatrix} $$

Perform the multiplication:

• First row, first column: $(1 \cdot 1 + 2 \cdot 3) = 7$
• First row, second column: $(1 \cdot 2 + 2 \cdot 1) = 4$
• Second row, first column: $(3 \cdot 1 + 1 \cdot 3) = 6$
• Second row, second column: $(3 \cdot 2 + 1 \cdot 1) = 7$

Thus, $$ A^2 = \begin{bmatrix} 7 & 4 \ 6 & 7 \end{bmatrix} $$

2. Calculate 2A

Now compute $2A$: $$ 2A = 2 \times \begin{bmatrix} 1 & 2 \ 3 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 2 \end{bmatrix} $$

3. Identify 5I

The identity matrix $I$ for a $2 \times 2$ matrix is: $$ I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} $$

Now calculate $5I$: $$ 5I = 5 \times \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} $$

4. Combine the results and check the equation

Substituting into the equation $A^2 - 2A - 5I$: $$ A^2 - 2A - 5I = \begin{bmatrix} 7 & 4 \ 6 & 7 \end{bmatrix} - \begin{bmatrix} 2 & 4 \ 6 & 2 \end{bmatrix} - \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} $$

Calculating step by step:

• Subtract $2A$: $$ \begin{bmatrix} 7 & 4 \ 6 & 7 \end{bmatrix} - \begin{bmatrix} 2 & 4 \ 6 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} $$

• Now subtract $5I$: $$ \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} - \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} $$

Thus, $$ A^2 - 2A - 5I = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} = 0 $$