The matrix $$A^2 + 4A - 5I$$, where $$I$$ is identity matrix and $$A = \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix}$$, equals :

First, we are given the matrix $$ A = \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix} $$ and need to compute $$ A^2 + 4A - 5I $$, where $$ I $$ is the identity matrix.

Start by computing $$ A^2 $$. Since $$ A^2 = A \times A $$, multiply matrix A by itself:

$$ A^2 = \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix} $$

Calculate each element:

• Top-left element: $$ (1)(1) + (2)(4) = 1 + 8 = 9 $$
• Top-right element: $$ (1)(2) + (2)(-3) = 2 - 6 = -4 $$
• Bottom-left element: $$ (4)(1) + (-3)(4) = 4 - 12 = -8 $$
• Bottom-right element: $$ (4)(2) + (-3)(-3) = 8 + 9 = 17 $$

So, $$ A^2 = \begin{bmatrix} 9 & -4 \\ -8 & 17 \end{bmatrix} $$.

Next, compute $$ 4A $$. Multiply each element of A by 4:

$$ 4A = 4 \times \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix} = \begin{bmatrix} 4 & 8 \\ 16 & -12 \end{bmatrix} $$

Now, compute $$ 5I $$. The identity matrix $$ I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$, so:

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

Now, combine the matrices: $$ A^2 + 4A - 5I $$. Add $$ A^2 $$ and $$ 4A $$ first:

$$ A^2 + 4A = \begin{bmatrix} 9 & -4 \\ -8 & 17 \end{bmatrix} + \begin{bmatrix} 4 & 8 \\ 16 & -12 \end{bmatrix} = \begin{bmatrix} 9+4 & -4+8 \\ -8+16 & 17-12 \end{bmatrix} = \begin{bmatrix} 13 & 4 \\ 8 & 5 \end{bmatrix} $$

Then subtract $$ 5I $$:

$$ \begin{bmatrix} 13 & 4 \\ 8 & 5 \end{bmatrix} - \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} = \begin{bmatrix} 13-5 & 4-0 \\ 8-0 & 5-5 \end{bmatrix} = \begin{bmatrix} 8 & 4 \\ 8 & 0 \end{bmatrix} $$

So, $$ A^2 + 4A - 5I = \begin{bmatrix} 8 & 4 \\ 8 & 0 \end{bmatrix} $$.

Now, compare this result with the options. Factor out a common factor of 4 from the matrix:

$$ \begin{bmatrix} 8 & 4 \\ 8 & 0 \end{bmatrix} = 4 \times \begin{bmatrix} 2 & 1 \\ 2 & 0 \end{bmatrix} $$

Check the options:

• Option A: $$ 4 \begin{bmatrix} 2 & 1 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 8 & 4 \\ 8 & 0 \end{bmatrix} $$ — matches exactly.
• Option B: $$ 4 \begin{bmatrix} 0 & -1 \\ 2 & 2 \end{bmatrix} = \begin{bmatrix} 0 & -4 \\ 8 & 8 \end{bmatrix} $$ — does not match.
• Option C: $$ 32 \begin{bmatrix} 2 & 1 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 64 & 32 \\ 64 & 0 \end{bmatrix} $$ — does not match.
• Option D: $$ 32 \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 32 & 32 \\ 32 & 0 \end{bmatrix} $$ — does not match.

Hence, the correct answer is Option A.