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 subtraction operation \(A-C\) is not defined because the two matrices do not have the same dimensions.

Step 1: Compare the dimensions of the matrices

Matrix A is a \(3 \times 2\) matrix and matrix C is a \(2 \times 2\) matrix. The rule for matrix addition and subtraction requires both matrices to be of the same dimension, meaning they must have exactly the same number of rows and the same number of columns.

Step 2: Assess possibility of operation

Given the dimensions, it is not possible to perform the operation \(A-C\) since the two matrices do not have the same dimensions. Thus, the subtraction operation cannot be performed.

Key Concepts

Matrix Addition and Subtraction

Matrix addition and subtraction are fundamental operations in matrix algebra. They allow us to combine matrices in various ways, but there is a crucial requirement that must be fulfilled: the matrices must have the same dimensions. This means that each corresponding element of the matrices can be directly paired with the other.

For instance, if you have two matrices, say matrix \(A\) and matrix \(C\), both of which are defined as follows:

• Matrix \(A\) has dimensions \(3 \times 2\), meaning it has 3 rows and 2 columns.
• Matrix \(C\) might have a different dimension, say \(2 \times 2\).

If these matrices do not share the same dimension, as in this example, addition or subtraction cannot be performed. The corresponding elements need to exist in both matrices to be able to directly perform the operations. This is why our operation \(A-C\) is undefined in the given exercise. By ensuring the dimensions are compatible, you can carry out these operations without issues.

Matrix Dimensions

Matrix dimensions are a fundamental aspect of matrices that determine how they can interact in operations. Knowing the dimensions of a matrix gives you information about its shape and the number of elements it contains.

A typical notation for matrix dimensions is \(m \times n\), where:

• \(m\) denotes the number of rows.
• \(n\) denotes the number of columns.

Understanding dimensions helps in visualizing the matrix and making decisions about operations.

For example, the dimension of Matrix \(A\) given in the problem statement is \(3 \times 2\), meaning it has three rows and two columns. Matrix \(C\) is \(2 \times 2\), indicating two rows and two columns. Because their dimensions are not equal, direct operations that require the same dimensions, like addition and subtraction, cannot be performed.

Knowing the matrix dimensions also aids in understanding whether multiplication is possible, which involves other rules but equally relies on understanding the size and shape of the matrices involved.

Matrix Algebra

Matrix algebra involves various operations beyond simple addition and subtraction; it includes multiplication, the finding of determinants, inverses, and more. Each of these operations has its own set of rules and requirements for matrices.

Matrix multiplication, for example, doesn't require the same dimensions, but it does require the number of columns in the first matrix to equal the number of rows in the second. Understanding these operations gives tools to solve systems of equations, transform geometric data, and undertake complex computations in fields ranging from computer graphics to statistics.

• In matrix algebra, matrices can be seen as linear transformations, providing a way to work with vector spaces in a consistent manner.
• Matrix identities and zero matrices play roles similar to numbers 0 and 1 in regular algebra.

Each operation and rule in matrix algebra stems from mathematical definitions and the need to systematically handle data and transformations. This exercise illustrates the importance of knowing whether operations such as subtraction are defined by checking dimensions, which is a good entry point to explore other powerful tools in matrix algebra.