Problem 41 - College Algebra

Question

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows If an operation is not defined, state the reason. $$A=\left[\begin{array}{rr}4 & 0 \\\\-3 & 5 \\\0 & 1\end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\\\-2 & -2\end{array}\right] \quad C=\left[\begin{array}{rr}1 & -1 \\\\-1 & 1\end{array}\right]$$ $$A-C$$

Step-by-Step Solution

Answer

The matrix subtraction operation \(A - C\) is not defined.

Step 1: Check the dimensions of the matrices

The matrix \(A\) has 3 rows and 2 columns, so its dimensions are 3x2. The matrix \(C\) has 2 rows and 2 columns, its dimensions are 2x2.

Step 2: Attempt the matrix subtraction

Because the two matrices do not have the same dimensions, the subtraction operation \(A - C\) is undefined in matrix algebra. In other words, it's not possible to subtract a 3x2 matrix from a 2x2 matrix.

Key Concepts

Matrix Operations

Matrix operations encompass a variety of procedures, including addition, subtraction, multiplication, and division (in the form of inversion), performed on matrices. Matrix subtraction is one particular operation where two matrices of the same dimensions are subtracted element by element. For example, if we have two matrices \( X \) and \( Y \), the subtraction \( X - Y \) results in a new matrix where each element \( x_{ij} \) from \( X \) is subtracted by the corresponding element \( y_{ij} \) from \( Y \).

This operation only makes sense when both matrices have the identical number of rows and columns, which ensures that for each position in \( X \) there is a corresponding position in \( Y \). A crucial aspect of subtraction is that it is a direct element-to-element procedure. In contrast to multiplication, there is no notion of combining rows and columns; the operation is straightforward - subtract one number from the other in the same position.

Matrix Dimensions

The term 'matrix dimensions' refers to the number of rows and columns that a matrix contains, conventionally expressed as 'rows \( \times \) columns'. Dimensions are fundamental when performing matrix operations, as they often dictate whether or not certain operations can be performed between two matrices.

For instance, as highlighted in our original exercise, matrix \( A \) with dimensions \( 3 \times 2 \) has three rows and two columns, while matrix \( C \) with dimensions \( 2 \times 2 \) has two rows and two columns. This difference in dimensions is a crucial factor for determining the viability of matrix operations such as addition and subtraction. Understanding and checking dimensions is an essential step before proceeding with any matrix operations to ensure that they are defined for the matrices in question.

Undefined Matrix Operations

Certain matrix operations become 'undefined' when the conditions required for performing them aren’t met. This is particularly important for operations such as addition and subtraction, which as mentioned before, require matrices to have matching dimensions. An operation is considered undefined when there is no mathematical procedure to produce a result according to the rules of matrix algebra.

Taking the original exercise as a case study, matrix \( A \) and matrix \( C \) cannot be subtracted because they have different dimensions: \( A \) is a \( 3 \times 2 \) matrix while \( C \) is a \( 2 \times 2 \) matrix. Whenever you encounter matrices with different dimensions, remember that addition and subtraction simply can't be done. It's much like attempting to subtract a three-dimensional shape from a two-dimensional one; they inhabit different realms of space and can't be directly compared element-wise.