Problem 99 - 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. $$C A-D$$

Step-by-Step Solution

Answer

No, the operation \(C A - D\) is not possible due to dimension mismatch. The result of \(C A\) is \(3 \times 3\) and \(D\) is \(2 \times 2\).

Step 1: Analyzing Matrix Multiplication Requirements

Consider the multiplication \(C A\). The matrix \(C\) is \(3 \times 2\) and \(A\) is \(2 \times 3\). It is clear from the property of matrix multiplication that we can perform the operation \(C A\) because the number of columns of \(C\) matches the number of rows of \(A\), yielding a matrix of \(3 \times 3\) dimensions.

Step 2: Analyzing Matrix Subtraction Requirements

Consider the operation \(C A-D\). For this operation to be possible, the matrices \(C A\) and \(D\) must have the same dimensions to subtract. The matrix \(D\) is given to be of \(2 \times 2\) dimension while the result of \(C A\) is of \(3 \times 3\) dimension. Hence, it is clear that the operation \(C A-D\) is not possible due to the difference in dimensions between the two matrices.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where you multiply two matrices to get another matrix. For this to work, it's essential to match the dimensions:

• The number of columns in the first matrix must equal the number of rows in the second matrix. This alignment is crucial for matrix multiplication to be valid.
• The resulting matrix will have dimensions based on the rows of the first matrix and the columns of the second matrix. For example, multiplying a \( 3 \times 2 \) matrix with a \( 2 \times 3 \) matrix results in a \( 3 \times 3 \) matrix.

When multiplying matrices, each entry in the resulting matrix is computed as the dot product of the corresponding row of the first matrix and the column of the second matrix. This means you'll multiply corresponding elements and sum them up. Matrix multiplication is not commutative, which means \( AB eq BA \) in general.

Matrix Dimensions

Every matrix has a specific set of dimensions, determined by the number of rows and columns it contains. Understanding these dimensions is essential because they dictate what operations can be performed:

• A matrix is described as \( m \times n \) where \( m \) is the number of rows and \( n \) is the number of columns.
• The dimensions must "fit" the operation, especially in multiplication, where one matrix's columns must match the other’s rows.
• For addition and subtraction, the matrices involved must have identical dimensions, ensuring each element aligns perfectly with a corresponding element in the other matrix.

Thus, knowing and being able to quickly determine a matrix's dimensions is important to effectively perform and understand matrix operations.

Matrix Subtraction

Matrix subtraction is an operation that allows you to subtract one matrix from another, element by element. For this operation to be possible:

• Both matrices involved must have the exact same dimensions. This means each matrix must have the same number and arrangement of rows and columns.
• Each element of the first matrix is reduced by the corresponding element of the second matrix.

If the dimensions do not match, as is the case with the problem involving matrices \( C \cdot A \) (\( 3 \times 3 \)) and \( D \) (\( 2 \times 2 \)), subtraction cannot be performed. This is because you won’t have a corresponding element in one matrix to subtract from the other. Always ensure dimensions align before attempting to subtract matrices.