Find matrix B, if matrix `A= [(1,5),(1,2)]`, matrix `C= [(2),(1)] and AB= 3C`

Answer

To find matrix B given that \( AB = 3C \), where \( A = \begin{pmatrix} 1 & 5 \\ 1 & 2 \end{pmatrix} \) and \( C = \begin{pmatrix} 2 \\ 1 \end{pmatrix} \), we can follow these steps:

### Step 1: Determine the order of matrix B
Matrix A is a \( 2 \times 2 \) matrix, and matrix C is a \( 2 \times 1 \) matrix. Since \( AB = 3C \), we need to find the order of matrix B.

- The product \( AB \) will have the same number of rows as A and the same number of columns as B.
- Given that C is \( 2 \times 1 \), \( 3C \) will also be \( 2 \times 1 \).
- Therefore, the number of rows in B must be 2, and the number of columns must be 1. Thus, matrix B is \( 2 \times 1 \).

### Step 2: Define matrix B
Let matrix B be represented as:
\[ B = \begin{pmatrix} x \\ y \end{pmatrix} \]

### Step 3: Set up the equation \( AB = 3C \)
Now we can set up the equation:
\[ AB = \begin{pmatrix} 1 & 5 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \cdot x + 5 \cdot y \\ 1 \cdot x + 2 \cdot y \end{pmatrix} = \begin{pmatrix} x + 5y \\ x + 2y \end{pmatrix} \]
And we know that:
\[ 3C = 3 \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 6 \\ 3 \end{pmatrix} \]

### Step 4: Set up the system of equations
From the equation \( AB = 3C \), we have:
\[ \begin{pmatrix} x + 5y \\ x + 2y \end{pmatrix} = \begin{pmatrix} 6 \\ 3 \end{pmatrix} \]
This gives us two equations:
1. \( x + 5y = 6 \) (Equation 1)
2. \( x + 2y = 3 \) (Equation 2)

### Step 5: Solve the system of equations
We can solve these equations using the elimination method.

From Equation 2:
\[ x + 2y = 3 \implies x = 3 - 2y \]

Substituting \( x \) in Equation 1:
\[ (3 - 2y) + 5y = 6 \]
This simplifies to:
\[ 3 + 3y = 6 \]
Subtracting 3 from both sides:
\[ 3y = 3 \implies y = 1 \]

Now substitute \( y = 1 \) back into Equation 2 to find \( x \):
\[ x + 2(1) = 3 \implies x + 2 = 3 \implies x = 1 \]

### Step 6: Write the final matrix B
Thus, we have:
\[ B = \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \]

### Final Answer
The matrix B is:
\[ B = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \]