Problem 100 - 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. $$C B-D$$

Step-by-Step Solution

Answer

The operation \(C \times B - D\) is not possible due to incompatible dimensions of matrices for subtraction.

Step 1: Check if multiplication is possible

Look at the dimensions of matrices \(C\) and \(B\). Matrix \(C\) has dimensions \(3 \times 2\) and matrix \(B\) has dimensions \(2 \times 3\). For multiplication, the column number of the first matrix should be equal to the row number of the second matrix. Hence, \(C \times B\) is possible and the dimensions of the result will be \(3 \times 3\).

Step 2: Check if subtraction is possible

Let's evaluate the dimensions of \(C \times B\) and \(D\). As concluded from step1, \(C \times B\) has dimension \(3 \times 3\) and \(D\) has dimension \(2 \times 2\). For subtraction, those matrices should have the same dimensions which is not the case here. Hence, \(C \times B - D\) is not possible.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is fundamental when performing matrix operations. In essence, the dimensions of a matrix refer to the number of rows and columns it consists of, often denoted as 'm x n', where 'm' is the number of rows and 'n' is the number of columns.

When it comes to matrix multiplication, for instance, the inner dimensions must match. That means if the first matrix is of size 'a x b' and the second matrix is of size 'c x d', for successful multiplication, 'b' and 'c' must be equal. The resulting matrix will have dimensions 'a x d'.

In the exercise provided, it's critical to recognize that while matrices A and B share the same dimensions, the operation in question cannot be completed due to the misalignment of dimensions between matrices C, B, and D. Matrix C (3x2) can be multiplied by matrix B (2x3), resulting in a new matrix with dimensions (3x3). However, this resultant matrix cannot then be subtracted from matrix D (2x2) because their dimensions are not the same.

Matrix Operations

Matrix operations such as addition, subtraction, and multiplication follow specific rules which are necessary to produce accurate results.

Matrix Multiplication

For instance, the multiplication process is not a matter of multiplying corresponding elements. Instead, it involves taking the row elements of the first matrix and multiplying them by the corresponding column elements of the second matrix, then adding the products to get each element of the resulting matrix.

Matrix Subtraction

On the other hand, subtraction (and addition) is more straightforward, with corresponding elements of equal-dimension matrices being subtracted (or added) to generate the resultant matrix. This means that in order to subtract one matrix from another, both must be of identical dimensions—a 3x3 matrix can only be subtracted from another 3x3 matrix, and hence the proposed operation in the exercise, subtracting a 2x2 matrix from a 3x3 matrix, is not permitted by the rules of matrix operations.

Matrices in Algebra

Matrices offer a powerful tool in algebra for organizing and simplifying complex calculations, especially when dealing with systems of linear equations, transformations, and vector calculations. They allow for compact representation of many values or coefficients and can be manipulated using matrix operations to solve for unknowns.

The application of matrices to solve algebraic problems can lead to quick and efficient solutions where traditional methods might be cumbersome. This requires an understanding of how to perform operations with matrices properly, as well as knowledge of when certain operations are possible based on the dimensions of the matrices involved.

For example, in the provided exercise, if matrix C multiplied by matrix B had resulted in a 2x2 matrix, then subtracting matrix D would have been a valid operation, possibly representing the difference between two transformations or the solution to a system of equations. However, since the operation yields a matrix of different dimensions, the algebraic operation cannot be completed, akin to attempting to solve a system of equations where the number of equations does not match the number of unknowns.