To find the product of the two given matrices:
[tex]\[
\left[\begin{array}{ll}
-4 & 4
\end{array}\right]
\][/tex]
and
[tex]\[
\left[\begin{array}{ccc}
-2 & -9 & 7 \\
4 & 8 & 3
\end{array}\right],
\][/tex]
we need to perform matrix multiplication. Let’s denote the first matrix as [tex]\( A \)[/tex] and the second matrix as [tex]\( B \)[/tex].
Matrix [tex]\( A \)[/tex]:
[tex]\[
A = \left[\begin{array}{ll}
-4 & 4
\end{array}\right]
\][/tex]
Matrix [tex]\( B \)[/tex]:
[tex]\[
B = \left[\begin{array}{ccc}
-2 & -9 & 7 \\
4 & 8 & 3
\end{array}\right]
\][/tex]
In matrix multiplication, the element of the resulting matrix at position [tex]\( (i, j) \)[/tex] is the dot product of the [tex]\( i \)[/tex]-th row of the first matrix and the [tex]\( j \)[/tex]-th column of the second matrix.
The resulting matrix will be a [tex]\( 1 \times 3 \)[/tex] matrix because the first matrix is [tex]\( 1 \times 2 \)[/tex] and the second matrix is [tex]\( 2 \times 3 \)[/tex].
Let’s calculate each element of the resulting matrix step-by-step.
### First Column
[tex]\[
\text{Element} (1,1) = (-4) \cdot (-2) + 4 \cdot 4
\][/tex]
[tex]\[
= 8 + 16 = 24
\][/tex]
### Second Column
[tex]\[
\text{Element} (1,2) = (-4) \cdot (-9) + 4 \cdot 8
\][/tex]
[tex]\[
= 36 + 32 = 68
\][/tex]
### Third Column
[tex]\[
\text{Element} (1,3) = (-4) \cdot 7 + 4 \cdot 3
\][/tex]
[tex]\[
= -28 + 12 = -16
\][/tex]
Thus, the resulting product matrix is:
[tex]\[
\left[\begin{array}{ccc}
24 & 68 & -16
\end{array}\right]
\][/tex]
In the problem, you were specifically asked to identify the element represented by [tex]\([?]\)[/tex]. As we've observed in our computations, [tex]\([?]\)[/tex] corresponds to the second column element, which is 68.
So, the complete product matrix and the specific element we are interested in are:
[tex]\[
\left[\begin{array}{ccc}
24 & {[68]} & -16
\end{array}\right]
\][/tex]
Hence, the element you're looking for, [tex]\([?]\)[/tex], is 68.
