Understanding the Concept:
Matrix multiplication is performed row-wise and column-wise.
If
\[
A=[a_{ij}]
\quad \text{and} \quad
B=[b_{ij}],
\]
then each entry of \(AB\) is obtained by multiplying the corresponding row of \(A\) with the corresponding column of \(B\).
Step 1: Write the matrices clearly.
\[
A=
\begin{bmatrix}
1 & 2
3 & 4
\end{bmatrix}
\]
and
\[
B=
\begin{bmatrix}
2 & 0
1 & 5
\end{bmatrix}
\]
We compute:
\[
AB
=
\begin{bmatrix}
1 & 2
3 & 4
\end{bmatrix}
\begin{bmatrix}
2 & 0
1 & 5
\end{bmatrix}
\]
Step 2: Find the first row entries.
First row, first column:
\[
(1)(2)+(2)(1)
=
2+2
=
4
\]
First row, second column:
\[
(1)(0)+(2)(5)
=
0+10
=
10
\]
Thus first row becomes:
\[
[4 \quad 10]
\]
Step 3: Find the second row entries.
Second row, first column:
\[
(3)(2)+(4)(1)
=
6+4
=
10
\]
Second row, second column:
\[
(3)(0)+(4)(5)
=
0+20
=
20
\]
Thus second row becomes:
\[
[10 \quad 20]
\]
Step 4: Write the final matrix.
Hence,
\[
AB=
\begin{bmatrix}
4 & 10
10 & 20
\end{bmatrix}
\]
Therefore,
\[
\boxed{
AB=
\begin{bmatrix}
4 & 10
10 & 20
\end{bmatrix}
}
\]