Let's solve the given matrix expression step-by-step.
We have the expression:
[tex]\[
\left[\begin{array}{cr}1 & -2 \\ -3 & 4 \\ 5 & 3\end{array}\right] + 2\left[\begin{array}{ll}0 & 1 \\ 6 & 7 \\ 5 & 0\end{array}\right]
\][/tex]
First, let's define the matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
A = \left[\begin{array}{cr}1 & -2 \\ -3 & 4 \\ 5 & 3\end{array}\right]
\][/tex]
[tex]\[
B = \left[\begin{array}{ll}0 & 1 \\ 6 & 7 \\ 5 & 0\end{array}\right]
\][/tex]
Next, we need to perform scalar multiplication on matrix [tex]\(B\)[/tex] with the scalar [tex]\(2\)[/tex]:
[tex]\[
2 \times B = 2 \times \left[\begin{array}{ll}0 & 1 \\ 6 & 7 \\ 5 & 0\end{array}\right] = \left[\begin{array}{ll}2 \times 0 & 2 \times 1 \\ 2 \times 6 & 2 \times 7 \\ 2 \times 5 & 2 \times 0\end{array}\right] = \left[\begin{array}{ll}0 & 2 \\ 12 & 14 \\ 10 & 0\end{array}\right]
\][/tex]
So, multiplying matrix [tex]\(B\)[/tex] by [tex]\(2\)[/tex] gives us:
[tex]\[
2B = \left[\begin{array}{ll}0 & 2 \\ 12 & 14 \\ 10 & 0\end{array}\right]
\][/tex]
Now, we need to add the matrices [tex]\(A\)[/tex] and [tex]\(2B\)[/tex]:
[tex]\[
A + 2B = \left[\begin{array}{cr}1 & -2 \\ -3 & 4 \\ 5 & 3\end{array}\right] + \left[\begin{array}{ll}0 & 2 \\ 12 & 14 \\ 10 & 0\end{array}\right]
\][/tex]
We perform the addition element-wise:
[tex]\[
\left[\begin{array}{cr}1 + 0 & -2 + 2 \\ -3 + 12 & 4 + 14 \\ 5 + 10 & 3 + 0\end{array}\right]
\][/tex]
Calculating each element, we get:
[tex]\[
\left[\begin{array}{cr}1 & 0 \\ 9 & 18 \\ 15 & 3\end{array}\right]
\][/tex]
Thus, the result of the matrix expression is:
[tex]\[
\left[\begin{array}{cr}1 & 0 \\ 9 & 18 \\ 15 & 3\end{array}\right]
\][/tex]
