The product of matrix B and C is matrix D begin{bmatrix}2 & -1&4 g & 0&32&h&0 end{bmatrix} times begin{bmatrix}-1 & 5 4&f-3&1 end{bmatrix}=begin{bmatr

Step 1 We are given the two matrices namely $B=2\left[\begin{array}{ccc}2& -1& 4\\ g& 0& 3\\ 2& h& 0\end{array}\right]\text{ and }C=\left[\begin{array}{cc}-1& 5\\ 4& f\\ -3& 1\end{array}\right]$
Using the basic rules of multiplication, we have
$B\times C=2\left[\begin{array}{ccc}2& -1& 4\\ g& 0& 3\\ 2& h& 0\end{array}\right]\times \left[\begin{array}{cc}-1& 5\\ 4& f\\ -3& 1\end{array}\right]$
$\Rightarrow B\times C=\left[\begin{array}{ccc}4& -2& 8\\ 2g& 0& 6\\ 4& 4h& 0\end{array}\right]\times \left[\begin{array}{cc}-1& 5\\ 4& f\\ -3& 1\end{array}\right]$
$\Rightarrow B\times C=\left[\begin{array}{cc}-4-8-24& 20-2f+8\\ -2g+0-18& 10g+6\\ -4+8h+0& 20+2hf+0\end{array}\right]$
Step 2
Now, we have $B\times C=\left[\begin{array}{cc}-4-8-24& 20-2f+8\\ -2g+0-18& 10g+6\\ -4+8h+0& 20+2hf+0\end{array}\right]$ . We are also given that $B\times C=\left[\begin{array}{cc}i& 24\\ -16& -4\\ 4& e\end{array}\right]$
Therefore, we now have
$B\times C=\left[\begin{array}{cc}-4-8-24& 20-2f+8\\ -2g+0-18& 10g+6\\ -4+8h+0& 20+2hf+0\end{array}\right]=\left[\begin{array}{cc}i& 24\\ -16& -4\\ 4& e\end{array}\right]$. Comparing the entries of the matrices, we have
-36=i, 28-2f=24
2g+18=16,10g+6=-4
8h-4=4,20+2hf=e
Solving these equations, we get
i=-36,f=2,g=-1,h=1,e=24
Step 3
(3). Answer: The value of e is 24.
(4). Answer: The value of g is -1.
(5). Answer: The value of f is 2.