(8 5           (5 9
12 16    *    3 7)
0 13
5 0)

Matrices calculate

To multiply the given matrices, we need to follow the standard matrix multiplication procedure. Here are the matrices you provided:

\[
A = \begin{pmatrix} 8 & 5 \\ 12 & 16 \\ 0 & 13 \\ 5 & 0 \end{pmatrix}
\quad \text{and} \quad
B = \begin{pmatrix} 5 & 9 \\ 3 & 7 \end{pmatrix}
\]

### Step-by-Step Multiplication:

Matrix \(A\) is a \(4 \times 2\) matrix, and matrix \(B\) is a \(2 \times 2\) matrix. The resulting matrix \(C\) will have dimensions \(4 \times 2\).

\[
C = AB = \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \\ c_{31} & c_{32} \\ c_{41} & c_{42} \end{pmatrix}
\]

Each element \(c_{ij}\) of matrix \(C\) is calculated as:

\[
c_{ij} = \sum_{k=1}^{2} a_{ik} \cdot b_{kj}
\]

**Calculating each element:**

1. **\(c_{11}\):**  
\[
c_{11} = (8 \times 5) + (5 \times 3) = 40 + 15 = 55
\]

2. **\(c_{12}\):**  
\[
c_{12} = (8 \times 9) + (5 \times 7) = 72 + 35 = 107
\]

3. **\(c_{21}\):**  
\[
c_{21} = (12 \times 5) + (16 \times 3) = 60 + 48 = 108
\]

4. **\(c_{22}\):**  
\[
c_{22} = (12 \times 9) + (16 \times 7) = 108 + 112 = 220
\]

5. **\(c_{31}\):**  
\[
c_{31} = (0 \times 5) + (13 \times 3) = 0 + 39 = 39
\]

6. **\(c_{32}\):**  
\[
c_{32} = (0 \times 9) + (13 \times 7) = 0 + 91 = 91
\]

7. **\(c_{41}\):**  
\[
c_{41} = (5 \times 5) + (0 \times 3) = 25 + 0 = 25
\]

8. **\(c_{42}\):**  
\[
c_{42} = (5 \times 9) + (0 \times 7) = 45 + 0 = 45
\]

### Final Matrix \(C\):

\[
C = \begin{pmatrix}
55 & 107 \\
108 & 220 \\
39 & 91 \\
25 & 45
\end{pmatrix}
\]

Would you like further details on any of these steps?

---

### Related Questions:

1. What are the rules for matrix multiplication?
2. Can matrix multiplication be commutative?
3. How do you find the determinant of a matrix?
4. What is an identity matrix and its role in multiplication?
5. How do you compute the inverse of a matrix?
6. What are the applications of matrix multiplication in real life?
7. Can two matrices of different dimensions be multiplied?
8. How does matrix multiplication differ from element-wise multiplication?

### Tip:

When multiplying matrices, ensure that the number of columns in the first matrix matches the number of rows in the second matrix.

Explore the step-by-step process of multiplying matrices, specifically a 4x2 matrix by a 2x2 matrix. Learn how to compute each element of the resulting matrix using the matrix multiplication formula. Suitable for high school students and anyone interested in matrix algebra.

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