Compute the product. This exercise should be done two ways: by hand and by using technology where possible. (If an answer is undefined, enter UNDEFINED in any cell of the matrix.)

	6	0	−4	


−4	
1
6

We are asked to compute the product of two matrices. The first matrix is a row matrix, and the second matrix is a column matrix:

\[
\text{Row matrix} = [6 \quad 0 \quad -4]
\]

\[
\text{Column matrix} =
\begin{bmatrix}
-4 \\
1 \\
6
\end{bmatrix}
\]

### Step 1: Matrix multiplication by hand

Matrix multiplication for a row matrix \(1 \times 3\) and a column matrix \(3 \times 1\) results in a single scalar (a \(1 \times 1\) matrix). To compute this, we multiply the corresponding elements and sum the results:

\[
\text{Product} = (6 \times -4) + (0 \times 1) + (-4 \times 6)
\]

Calculating each term:

- \(6 \times -4 = -24\)
- \(0 \times 1 = 0\)
- \(-4 \times 6 = -24\)

Summing these values:

\[
-24 + 0 - 24 = -48
\]

Thus, the product is:

\[
\boxed{-48}
\]

### Step 2: Matrix multiplication using technology

Using a calculator or software like Python (NumPy), the matrix multiplication will also give:

\[
\boxed{-48}
\]

Thus, whether done by hand or using technology, the result is \(-48\).

---

Do you want further details on matrix multiplication or any clarification on this problem?

### Here are 5 related questions:
1. What is the general rule for multiplying two matrices?
2. How does matrix multiplication differ from element-wise multiplication?
3. Can you multiply a \(2 \times 3\) matrix with a \(3 \times 1\) matrix?
4. What are the conditions for a matrix product to be defined?
5. How can you interpret the result of multiplying a row vector by a column vector?

**Tip:** Always check matrix dimensions before attempting multiplication to ensure the operation is defined.

Compute the product. This exercise should be done two ways: by hand and by using technology where possible. (If an answer is undefined, enter UNDEFINED in any cell of the matrix.) 6 0 −4, −4 1 6

This tutorial demonstrates how to compute the matrix product of a row matrix [6 0 -4] and a column matrix [-4 1 6], both by hand and using technology. The step-by-step solution includes detailed explanations of matrix multiplication concepts and relevant formulas, suitable for Grades 10-12 students.

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