- Analyze dimensions: Multiply a \(1 \times 2\) matrix by a \(2 \times 2\) matrix to yield a \(1 \times 2\) matrix.
- Perform multiplication: Calculate the resulting elements as \((4 \times -2) + (2 \times 7) = 6\) and \((4 \times 5) + (2 \times -1) = 18\).
- Finalize: The resulting matrix is \(\boxed{\begin{bmatrix} 6 & 18 \end{bmatrix}}\).
Explanation
- Analysis of Matrix Dimensions
To find the product of the given matrices \(\begin{bmatrix} 4 & 2 \end{bmatrix} \times \begin{bmatrix} -2 & 5 \ 7 & -1 \end{bmatrix}\), we first analyze the dimensions of the matrices. The first matrix is a \(1 \times 2\) row vector, and the second matrix is a \(2 \times 2\) square matrix. The product of a \(1 \times 2\) matrix and a \(2 \times 2\) matrix results in a new \(1 \times 2\) matrix. We denote the product matrix as \(\begin{bmatrix} c_{11} & c_{12} \end{bmatrix}\).
- Performing the Matrix Multiplication
Matrix multiplication follows the rule that the element at row \(i\), column \(j\) is calculated by taking the dot product of the \(i\)-th row of the first matrix and the \(j\)-th column of the second matrix.
For the element \(c_{11}\):
\[c_{11} = (4 \times -2) + (2 \times 7) = -8 + 14 = 6.\]
For the element \(c_{12}\):
\[c_{12} = (4 \times 5) + (2 \times -1) = 20 - 2 = 18.\]
Combining these, we get the resulting matrix \(\begin{bmatrix} 6 & 18 \end{bmatrix}\).
- Conclusion
Comparing our result \(\begin{bmatrix} 6 & 18 \end{bmatrix}\) with the provided options, we can see that it matches the last option. Therefore, the product of the given matrices is \(\boxed{\begin{bmatrix} 6 & 18 \end{bmatrix}}\).
Examples
Matrix multiplication is a fundamental tool in various fields, including computer graphics, economics, and physics. For example, in computer graphics, a matrix can represent a linear transformation like scaling, rotation, or translation of a point in space. By multiplying a vector (representing a point) by a transformation matrix, we can efficiently compute the new position of that point, enabling animations and 3D rendering in video games and movies.