Problem 62 - College Algebra and Calculus: An Applied Approach

Question

Let matrices \(A, B, C\), and \(D\) be of orders \(2 \times 3,2 \times 3,3 \times 2\), and \(2 \times 2\), respectively. Determine whether the matrices are of proper order to perform the operation(s). If so, give the order of the answer. $$ B C $$

Step-by-Step Solution

Answer

Yes, the multiplication operation can be performed on matrices B and C since their orders are agreeable for multiplication. The order of the resulting matrix will be 2x2.

Step 1: Verify the Conditions for Matrix Multiplication

Matrix B is of order 2x3 and matrix C is of order 3x2. Ensure that the number of columns in matrix B is equal to the number of rows in matrix C, which is the condition for matrix multiplication. Here the number of columns in B is 3 and the number of rows in C is also 3.

Step 2: Determine the Order of the Resulting Matrix

After the condition is verified, determine the size of the resulting matrix. The order of the resulting matrix will be formed by the number of rows from the first matrix and number of columns from the second matrix. Therefore, the order of the matrix BC is equivalent to the number of rows in B and the number of columns in C, which is 2x2.

Key Concepts

Matrix Order

The term "matrix order" refers to the dimensions of a matrix, describing how many rows and columns it has. Matrix order is expressed in the form "rows by columns" or \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. This terminology is essential when performing matrix operations, especially multiplication, as a matrix must have a specific order to be compatible with another matrix.

• For matrix \(B\), the order is \(2 \times 3\): 2 rows and 3 columns.
• For matrix \(C\), the order is \(3 \times 2\): 3 rows and 2 columns.
• Orders determine if two matrices can be multiplied.

Properly understanding and identifying matrix order helps in determining whether certain operations, like multiplication, can be carried out. This simplifies mathematical calculations and avoids errors.

Matrix Dimensions

"Matrix dimensions" refer to the size of a matrix, described quantitatively by its rows and columns. Knowing the dimensions is crucial when determining compatibility for performing matrix operations.
For a successful multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. Therefore, understanding matrix dimensions is integral to solving matrix problems accurately. Here are some key points:

• Matrix \(B\) has 3 columns, and matrix \(C\) has 3 rows.
• This alignment is perfect for multiplication since \(3 = 3\).
• If matrix \(B\) had more or fewer columns, or if matrix \(C\) had different rows, multiplication wouldn't be possible.

Thus, matrix dimensions are fundamental in ensuring matrices line up correctly for operations like multiplication.

Matrix Operations

"Matrix operations" encompass various techniques to combine or manipulate matrices, with multiplication being among the most common. In matrix multiplication, verifying the alignment of dimensions is crucial, as previously discussed. Here’s how the operations usually pan out:

• First, check that the number of columns in the first matrix matches the number of rows in the second.
• Multiply corresponding elements, then sum them to get each element of the resulting matrix.
• The resulting matrix \(BC\) will take on new dimensions, dictated by the rows of the first matrix and columns of the second – in this case, a \(2 \times 2\) matrix.

Understanding how these operations work not only aids in correctly solving mathematical problems but also in appreciating complex algorithms in computer science. Matrix operations allow for efficient data manipulation and are foundational in fields like machine learning.