Given the matrices:
[tex]A=\begin{bmatrix}{10} & {4} & {0} \\ {1} & {3} & {1}\end{bmatrix},B=\begin{bmatrix}{4} & {1} & \\ {2} & {2} & {} \\ {0} & {-1} & \end{bmatrix}[/tex]
we will find the value of AB + I
First, we will find the product of AB as follows:
[tex]AB=\begin{bmatrix}{10} & {4} & {0} \\ {1} & {3} & {1}\end{bmatrix}\cdot\begin{bmatrix}{4} & {1} & \\ {2} & {2} & {} \\ {0} & {-1} & \end{bmatrix}=\begin{bmatrix}{10\cdot4+2\cdot4+0\cdot0} & {1\cdot01+4\cdot2+0\cdot-1} & {} \\ {1\cdot4+3\cdot2+1\cdot0} & {1\cdot1+3\cdot2+1\cdot-1} & {} \\ {} & {} & {}\end{bmatrix}[/tex]
simplifying the answer:
[tex]AB=\begin{bmatrix}{48} & {18} & {} \\ {10} & {6} & {} \\ {} & {} & {}\end{bmatrix}[/tex]
Now, we will add the unity matrix to the answer:
[tex]AB+I=\begin{bmatrix}{48} & {18} & {} \\ {10} & {6} & {} \\ {} & {} & {}\end{bmatrix}+\begin{bmatrix}{1} & {0} & {} \\ {0} & {1} & {} \\ {} & {} & {}\end{bmatrix}=\begin{bmatrix}{49} & {18} & {} \\ {10} & {7} & {} \\ {} & {} & {}\end{bmatrix}[/tex]
So, the answer will be option D
