Questions: Find the following matrix product, if it exists. [ left[beginarrayrr -3 9 -5 -7 endarrayright]left[beginarrayl 8 7 endarrayright] ] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The matrix product is B. The product does not exist.

Find the following matrix product, if it exists.

[
left[beginarrayrr
-3 9
-5 -7
endarrayright]left[beginarrayl
8
7
endarrayright]
]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The matrix product is
B. The product does not exist.

Transcript text: Find the following matrix product, if it exists. \[ \left[\begin{array}{rr} -3 & 9 \\ -5 & -7 \end{array}\right]\left[\begin{array}{l} 8 \\ 7 \end{array}\right] \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\left[\begin{array}{rr}-3 & 9 \\ -5 & -7\end{array}\right]\left[\begin{array}{l}8 \\ 7\end{array}\right]=\square$ (Simplify your answer.) B. The product does not exist.

Solution

Solution Steps

Step 1: Check for Compatibility

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, Matrix A has 2 columns and Matrix B has 2 rows, which are compatible for multiplication.

Step 2: Initialize the Product Matrix

The product matrix C will have dimensions 2x1.

Step 3: Calculate the Product

For each element $c_{ij}$ in matrix $C$, calculate the sum of the products of the corresponding elements from the $i$th row of $A$ and the $j$th column of $B$. Formally, $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$ for all $i = 1, 2, ..., m$ and $j = 1, 2, ..., p$. Where $n$ is the number of columns in $A$ (or rows in $B$), $m$ is the number of rows in $A$, and $p$ is the number of columns in $B$.

Final Answer:

The product of matrices A and B is: [39] [-89]

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