Compute and simplify the matrix product given below, if possible. To receive full credit, you
must show all intermediate steps. If the matrix product is not possible, you must explain why
not.

7 −3 h
5 g −1



2 k
−5 9
8 −6

 =

We are given two matrices and asked to compute the product of the two, if possible. Let’s first write them down clearly:

Matrix \( A \) is:
\[
A = \begin{pmatrix} 
7 & -3 & h \\
5 & g & -1 
\end{pmatrix}
\]

Matrix \( B \) is:
\[
B = \begin{pmatrix}
2 & k \\
-5 & 9 \\
8 & -6
\end{pmatrix}
\]

### Step 1: Determine if the product is possible.
- Matrix \( A \) is a \( 2 \times 3 \) matrix (2 rows, 3 columns).
- Matrix \( B \) is a \( 3 \times 2 \) matrix (3 rows, 2 columns).

Matrix multiplication is only possible when the number of columns in the first matrix (Matrix \( A \)) is equal to the number of rows in the second matrix (Matrix \( B \)). Since both matrices meet this condition (3 columns in \( A \) and 3 rows in \( B \)), the product is possible.

The result of multiplying a \( 2 \times 3 \) matrix by a \( 3 \times 2 \) matrix will be a \( 2 \times 2 \) matrix.

### Step 2: Matrix multiplication formula.
To multiply two matrices, we use the following general formula:
\[
C = A \times B
\]
Where \( C \) is the resulting matrix, and each element \( C_{ij} \) of the resulting matrix is calculated as:
\[
C_{ij} = \sum_{k=1}^{n} A_{ik} \times B_{kj}
\]
This means that the element in row \( i \) and column \( j \) of the resulting matrix is the dot product of the \( i \)-th row of matrix \( A \) with the \( j \)-th column of matrix \( B \).

### Step 3: Perform the matrix multiplication.
We will now compute each element of the resulting \( 2 \times 2 \) matrix:

1. **First row, first column**:
\[
C_{11} = (7 \times 2) + (-3 \times -5) + (h \times 8) = 14 + 15 + 8h = 29 + 8h
\]

2. **First row, second column**:
\[
C_{12} = (7 \times k) + (-3 \times 9) + (h \times -6) = 7k - 27 - 6h = 7k - 27 - 6h
\]

3. **Second row, first column**:
\[
C_{21} = (5 \times 2) + (g \times -5) + (-1 \times 8) = 10 - 5g - 8 = 2 - 5g
\]

4. **Second row, second column**:
\[
C_{22} = (5 \times k) + (g \times 9) + (-1 \times -6) = 5k + 9g + 6 = 5k + 9g + 6
\]

### Step 4: Write the resulting matrix.
The resulting matrix product is:
\[
C = \begin{pmatrix}
29 + 8h & 7k - 27 - 6h \\
2 - 5g & 5k + 9g + 6
\end{pmatrix}
\]

This is the simplified form of the matrix product.

Would you like a more detailed breakdown of any of these steps?

---

**Related questions**:
1. How do we handle matrix multiplication if both matrices are square?
2. What are the conditions under which matrix multiplication is undefined?
3. What are the rules for the multiplication of identity matrices?
4. How does matrix multiplication differ from element-wise multiplication?
5. Can you perform a scalar multiplication on a matrix and how does it affect the matrix?

**Tip:** Always ensure the number of columns in the first matrix matches the number of rows in the second matrix when performing matrix multiplication.

Compute and simplify the matrix product given below, if possible. To receive full credit, you must show all intermediate steps. If the matrix product is not possible, you must explain why not.

7 −3 h
5 g −1



2 k
−5 9
8 −6

 = 

Explore the step-by-step process of multiplying a 2x3 matrix by a 3x2 matrix. This matrix multiplication problem demonstrates key concepts in linear algebra, including matrix product rules and detailed calculations of matrix elements. Suitable for Grades 11-12 or college students.

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