Determine whether the product is defined. If it is defined,...

Determine whether the product is defined. If it is defined, find the product; if it is not, say "not defined."
$$
\left[\begin{array}{lll}
1 & 0 & 1 \\
2 & 4 & 1 \\
3 & 6 & 1
\end{array}\right]\left[\begin{array}{rr}
1 & 3 \\
6 & 2 \\
8 & -1
\end{array}\right]
$$

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Key Concepts

Dot Product Calculation

The dot product calculation in matrix multiplication involves multiplying corresponding elements of the rows and columns from the two matrices and then summing those products. This operation is fundamental for computing each entry in the resultant matrix and encapsulates both scalar multiplication and addition.

Dimension Compatibility

Matrix multiplication is only defined when the number of columns of the first matrix is equal to the number of rows of the second matrix. This compatibility condition ensures that each element in the product matrix can be computed by taking the dot product of a row from the first matrix and a column from the second matrix.

Matrix Multiplication

Matrix multiplication is a binary operation in which two matrices are combined to produce a third matrix. This operation is performed by taking the dot product of rows from the first matrix with columns from the second matrix, thereby summing the products of their corresponding entries. The resulting matrix has dimensions determined by the number of rows in the first matrix and the number of columns in the second matrix.

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