If `((1,2,3))A=((4,5))`, what is the order of matrix A?

Options

A. `3xx2`
B. `3xx1`
C. `2xx3`
D. `2xx1`

Answer

To determine the order of matrix \( A \) in the equation \( (1, 2, 3)A = (4, 5) \), we can follow these steps:

### Step 1: Identify the order of the first matrix
The first matrix \( (1, 2, 3) \) is a row matrix with 1 row and 3 columns. Therefore, its order is \( 1 \times 3 \).

### Step 2: Identify the order of the second matrix
The second matrix \( (4, 5) \) is also a row matrix with 1 row and 2 columns. Thus, its order is \( 1 \times 2 \).

### Step 3: Set up the equation for matrix multiplication
In the equation \( (1, 2, 3)A = (4, 5) \), we denote the order of matrix \( A \) as \( P \times Q \). According to the rules of matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.

### Step 4: Apply the multiplication rule
From our first matrix \( (1, 2, 3) \), we have:
- Number of columns = 3

For matrix \( A \):
- Number of rows = \( P \)
- Number of columns = \( Q \)

Since the first matrix has 3 columns, we have:
\[ P = 3 \]

### Step 5: Determine the resulting matrix order
The resulting matrix \( (4, 5) \) has an order of \( 1 \times 2 \). Therefore, the product of the matrices \( (1, 2, 3) \) and \( A \) must also have the same order:
\[ 1 \times 2 \]

This means:
- The number of rows in the resulting matrix (which is the same as the number of rows in the first matrix) is 1.
- The number of columns in the resulting matrix (which is the same as the number of columns in matrix \( A \)) is 2.

Thus, we have:
\[ Q = 2 \]

### Step 6: Conclusion about the order of matrix \( A \)
Combining the values we found:
- \( P = 3 \)
- \( Q = 2 \)

The order of matrix \( A \) is:
\[ 3 \times 2 \]

### Final Answer
The order of matrix \( A \) is \( 3 \times 2 \).

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