If `A=[{:(1,3),(4,1):}]`, then find `|3A'|`.

Answer

To find \( |3A'| \), we will follow these steps:

### Step 1: Define the Matrix \( A \)
The given matrix \( A \) is:
\[ A = \begin{pmatrix} 1 & 3 \\ 4 & 1 \end{pmatrix} \]

### Step 2: Find the Transpose of Matrix \( A \)
The transpose of a matrix \( A \), denoted as \( A' \), is obtained by swapping the rows and columns. Thus,
\[ A' = \begin{pmatrix} 1 & 4 \\ 3 & 1 \end{pmatrix} \]

### Step 3: Calculate the Determinant of \( A' \)
The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula \( ad - bc \). For our matrix \( A' \):
\[ |A'| = (1 \cdot 1) - (4 \cdot 3) = 1 - 12 = -11 \]

### Step 4: Use the Property of Determinants
We know that if \( k \) is a constant and \( A \) is an \( n \times n \) matrix, then:
\[ |kA| = k^n |A| \]
In our case, \( k = 3 \) and \( n = 2 \) (since \( A' \) is a \( 2 \times 2 \) matrix). Therefore:
\[ |3A'| = 3^2 |A'| = 9 |A'| \]

### Step 5: Substitute the Determinant of \( A' \)
Now we substitute the value of \( |A'| \) we calculated in Step 3:
\[ |3A'| = 9 \cdot (-11) = -99 \]

### Final Answer
Thus, the value of \( |3A'| \) is:
\[ \boxed{-99} \]

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