Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

Yes, the multiplication of Matrix B and Matrix C is possible. The resultant matrix will have a dimension of \(2 \times 2\).

Step 1: Identification of Matrix Dimensions

The dimensions of the matrices to be multiplied are first identified. Matrix B is of dimension \(2 \times 3\) (2 rows and 3 columns) and Matrix C is of dimension \(3 \times 2\) (3 rows and 2 columns).

Step 2: Verification of Matrix Multiplication Conditions

The number of columns in the first matrix (B) should equal the number of rows in the second matrix (C) for the multiplication to be possible. In this case, Matrix B has 3 columns and Matrix C has 3 rows, so the matrices can be multiplied.

Step 3: Determination of Resultant Matrix Dimension

When two matrices can be multiplied, the dimensions of the resulting matrix should be equivalent to the number of rows in the first matrix and the number of columns in the second matrix. Thus, after the multiplication of Matrix B and Matrix C, which is allowed, the resultant matrix dimension will be \(2 \times 2\) (2 rows and 2 columns).

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in matrix operations. A matrix's dimension reflects the number of rows and columns it contains, expressed in the form of rows by columns, such as \(2 \times 3\). This format means the matrix has 2 rows and 3 columns. Recognizing these dimensions is the first and most vital step in performing matrix operations, especially multiplication.

For instance, in our exercise, Matrix B has dimensions of \(2 \times 3\), while Matrix C has dimensions of \(3 \times 2\). Each number reveals how the matrix is structured, which directly affects whether and how the matrices can be multiplied together. Understanding this foundational concept sets the stage for further matrix operations.

Matrix Operations

Matrix operations include a variety of calculations, with multiplication being one of the most common yet intricate. For two matrices to be multiplied, specific conditions must be met. The number of columns in the first matrix must match the number of rows in the second matrix. If this condition is satisfied, the matrices are conformable for multiplication.

• For instance, if Matrix B (\(2 \times 3\)) is multiplied by Matrix C (\(3 \times 2\)), the inner dimensions (3 from Matrix B and 3 from Matrix C) must align.
• This alignment enables each element from the row of the first matrix to be multiplied with the corresponding element from the column of the second matrix.

Meeting these criteria is essential for the matrices to engage in multiplication, leading us to calculate the resultant matrix.

Resultant Matrix Dimension

After verifying that two matrices can be multiplied, the next step is determining the dimension of the resultant matrix. The dimensions are derived from the outer dimensions — the number of rows from the first matrix and the number of columns from the second matrix.

For the multiplication of Matrix B (\(2 \times 3\)) and Matrix C (\(3 \times 2\)), the resultant matrix dimension is \(2 \times 2\).
This new matrix will have dimensions based on:

• The number of rows in Matrix B, which is 2
• The number of columns in Matrix C, which is 2

Therefore, the resulting matrix will be a square matrix with 2 rows and 2 columns. Understanding how these dimensions are calculated helps predict the size of the matrix resulting from multiplication, a critical aspect of mastering matrix basics.