# 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{bmatrix}i & 24 -16&-44&e end{bmatrix} 3.From the expression above, what should be the value of e? 4.From the expression above, what should be the value of g? 5.From the expression above, what should be the value of f?

Question
Matrices
The product of matrix B and C is matrix D
$$\begin{bmatrix}2 & -1&4 \\g & 0&3\\2&h&0 \end{bmatrix} \times \begin{bmatrix}-1 & 5 \\4&f\\-3&1 \end{bmatrix}=\begin{bmatrix}i & 24 \\-16&-4\\4&e \end{bmatrix}$$
3.From the expression above, what should be the value of e?
4.From the expression above, what should be the value of g?
5.From the expression above, what should be the value of f?

2021-02-25
Step 1 We are given the two matrices namely $$B=2\begin{bmatrix}2 & -1&4 \\g & 0&3\\2&h&0 \end{bmatrix} \text{ and } C=\begin{bmatrix}-1 & 5 \\4&f\\-3&1 \end{bmatrix}$$
Using the basic rules of multiplication, we have
$$B \times C=2\begin{bmatrix}2 & -1&4 \\g & 0&3\\2&h&0 \end{bmatrix} \times \begin{bmatrix}-1 & 5 \\4&f\\-3&1 \end{bmatrix}$$
$$\Rightarrow B \times C = \begin{bmatrix}4 & -2&8 \\2g & 0&6\\4&4h&0 \end{bmatrix} \times \begin{bmatrix}-1 & 5 \\4&f\\-3&1 \end{bmatrix}$$
$$\Rightarrow B \times C = \begin{bmatrix}-4-8-24 & 20-2f+8 \\-2g+0-18 & 10g+6\\-4+8h+0&20+2hf+0 \end{bmatrix}$$
Step 2
Now, we have $$B \times C = \begin{bmatrix}-4-8-24 & 20-2f+8 \\-2g+0-18 & 10g+6\\-4+8h+0&20+2hf+0 \end{bmatrix}$$ . We are also given that $$B \times C=\begin{bmatrix}i & 24 \\-16&-4\\4&e \end{bmatrix}$$
Therefore, we now have
$$B \times C=\begin{bmatrix}-4-8-24 & 20-2f+8 \\-2g+0-18 & 10g+6\\-4+8h+0&20+2hf+0 \end{bmatrix}=\begin{bmatrix}i & 24 \\-16&-4\\4&e \end{bmatrix}$$. 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.

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