Question

Find each product, if possible. [22−183270][8−1091202]\left[ \begin{matrix} \sqrt{2} & \sqrt{2} & -\sqrt{18} \ \sqrt{3} & \sqrt{27} & 0 \end{matrix} ight] \left[ \begin{matrix} 8 & -10 \ 9 & 12 \ 0 & 2 \end{matrix} ight]

Accepted Answer

Identify the dimensions of the two matrices: the first matrix is a 3x2 matrix (3 rows and 2 columns), and the second matrix is a 2x3 matrix (2 rows and 3 columns). Recall the rule for matrix multiplication: the number of columns in the first matrix must equal the number of rows in the second matrix. Here, the first matrix has 2 columns and the second has 2 rows, so multiplication is possible. Determine the dimensions of the resulting product matrix: it will have the number of rows of the first matrix and the number of columns of the second matrix, so the product will be a 3x3 matrix. To find each entry in the product matrix, multiply corresponding elements from the rows of the first matrix and the columns of the second matrix, then sum those products. For example, the entry in row i and column j of the product is calculated as:  $$\sum_{k=1}^{2} (a_{ik} 	imes b_{kj})$$ where $$a_{ik} is an $$element from the first matrix and $$b_{kj} is an $$element from the second matrix. Repeat this process for all entries in the 3x3 product matrix to complete the multiplication.

Find each product, if possible.
$[\begin{matrix} \sqrt{2} & \sqrt{2} & - \sqrt{18} \\ \sqrt{3} & \sqrt{27} & 0 \end{matrix}] [\begin{matrix} 8 & - 10 \\ 9 & 12 \\ 0 & 2 \end{matrix}]$

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Key Concepts

Matrix Dimensions and Compatibility

Matrix Multiplication Procedure

Resulting Matrix Size

Find each product, if possible. [22−183270][8−1091202]\(\left\)[ \(\begin{matrix}\) \(\sqrt{2}\) & \(\sqrt{2}\) & -\(\sqrt{18}\) \\ \(\sqrt{3}\) & \(\sqrt{27}\) & 0 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) 8 & -10 \\ 9 & 12 \\ 0 & 2 \(\end{matrix}\) \(\right\)]