2 1 a b c d 58914 tentukan nilai matriks 4 1 abcd nya

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2 1 a b c d = 5,8,9,14 tentukan nilai matriks 4 1 a,b,c,d nya

Asked Nov 20 at 03:54

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Basic Answer

Step 1: Set Up the Matrix Equation

Given the matrix equation:
$$\left(\begin{array}{cc}2& 1\\ 4& 1\end{array}\right)\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=\left(\begin{array}{cc}5& 8\\ 9& 14\end{array}\right)$$

Step 2: Multiply the Matrices

Perform the matrix multiplication:
$$\left(\begin{array}{cc}2a+c& 2b+d\\ 4a+c& 4b+d\end{array}\right)=\left(\begin{array}{cc}5& 8\\ 9& 14\end{array}\right)$$

Step 3: Set Up the System of Equations

From the matrix multiplication, we get the following system of equations:

1. $2a+c=5$
2. $2b+d=8$
3. $4a+c=9$
4. $4b+d=14$

Step 4: Solve the System of Equations

First, solve for $a$ and $c$ using equations 1 and 3:
$$2a+c=5\text{(Equation 1)}$$
$$4a+c=9\text{(Equation 3)}$$

Subtract Equation 1 from Equation 3:
$$(4a+c)-(2a+c)=9-5$$
$$2a=4$$
$$a=2$$

Substitute $a=2$ into Equation 1:
$$2(2)+c=5$$
$$4+c=5$$
$$c=1$$

Next, solve for $b$ and $d$ using equations 2 and 4:
$$2b+d=8\text{(Equation 2)}$$
$$4b+d=14\text{(Equation 4)}$$

Subtract Equation 2 from Equation 4:
$$(4b+d)-(2b+d)=14-8$$
$$2b=6$$
$$b=3$$

Substitute $b=3$ into Equation 2:
$$2(3)+d=8$$
$$6+d=8$$
$$d=2$$

Final Answer

The values of the matrix elements are:
$$a=2,b=3,c=1,d=2$$

Thus, the matrix is:
$$\left(\begin{array}{cc}2& 3\\ 1& 2\end{array}\right)$$