Find each product, if possible. See Examples 5–7. <1x2 Matrix>

Verified step by step guidance

1. Step 1: Understand the problem. We are asked to find the product of a matrix. Since the problem mentions a '1x2 Matrix', it implies we have a matrix with 1 row and 2 columns.
2. Step 2: Identify the matrices involved. A 1x2 matrix looks like this: \([a, b]\). To multiply it with another matrix, we need to know the dimensions of the second matrix.
3. Step 3: Check matrix multiplication compatibility. For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.
4. Step 4: Set up the multiplication. If the second matrix is a 2x1 matrix, for example \(\begin{bmatrix} c \\ d \end{bmatrix}\), then the multiplication is possible.
5. Step 5: Perform the multiplication. Multiply each element of the row of the first matrix by the corresponding element of the column of the second matrix and sum the results: \(a \cdot c + b \cdot d\).

Key Concepts

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce a third matrix. The number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix's dimensions are determined by the number of rows from the first matrix and the number of columns from the second matrix. Each element in the resulting matrix is calculated as the dot product of the corresponding row and column.

Matrix Dimensions

Matrix dimensions refer to the size of a matrix, expressed as 'rows x columns'. For example, a 1x2 matrix has 1 row and 2 columns. Understanding dimensions is crucial for determining whether two matrices can be multiplied together, as the inner dimensions must match. This concept is foundational in linear algebra and affects how matrices interact in operations.

Product of Matrices

The product of matrices is the result obtained from multiplying two matrices together. This operation is not commutative, meaning that the order of multiplication matters; A x B is not necessarily equal to B x A. The product can only be computed if the matrices meet the dimensional requirements, and the resulting matrix encapsulates the linear transformations represented by the original matrices.