Evaluate : `[{:(,4 sin 30^@ 2 cos 60^@), (,sin 90^@ 2 cos 0^@):}] [{:(,4,5),(,5,4):}]`

Answer

To evaluate the expression given in the question, we will follow these steps:

### Step 1: Identify the Values of Trigonometric Functions
We need to evaluate the trigonometric functions involved in the matrix:
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \sin 90^\circ = 1 \)
- \( \cos 0^\circ = 1 \)

### Step 2: Construct the First Matrix
Using the values from Step 1, we can construct the first matrix:
\[ \begin{pmatrix} 4 \sin 30^\circ & 2 \cos 60^\circ \\ \sin 90^\circ & 2 \cos 0^\circ \end{pmatrix} = \begin{pmatrix} 4 \cdot \frac{1}{2} & 2 \cdot \frac{1}{2} \\ 1 & 2 \cdot 1 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \]

### Step 3: Write the Second Matrix
The second matrix is given as:
\[ \begin{pmatrix} 4 & 5 \\ 5 & 4 \end{pmatrix} \]

### Step 4: Multiply the Two Matrices
Now we will multiply the two matrices:
\[ \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 4 & 5 \\ 5 & 4 \end{pmatrix} \]

To perform the multiplication, we will calculate each element of the resulting matrix:

- **First Row, First Column**:
\[ (2 \cdot 4) + (1 \cdot 5) = 8 + 5 = 13 \]

- **First Row, Second Column**:
\[ (2 \cdot 5) + (1 \cdot 4) = 10 + 4 = 14 \]

- **Second Row, First Column**:
\[ (1 \cdot 4) + (2 \cdot 5) = 4 + 10 = 14 \]

- **Second Row, Second Column**:
\[ (1 \cdot 5) + (2 \cdot 4) = 5 + 8 = 13 \]

### Step 5: Write the Resulting Matrix
Combining all the calculated elements, we get the resulting matrix:
\[ \begin{pmatrix} 13 & 14 \\ 14 & 13 \end{pmatrix} \]

### Final Answer
Thus, the evaluated result of the given matrix expression is:
\[ \begin{pmatrix} 13 & 14 \\ 14 & 13 \end{pmatrix} \]

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