Problem 41 - Precalculus: Mathematics for Calculus

Question

The matrices \(A, B,\) and \(C\) are defined as follows. $$A=\left[\begin{array}{rrr}0.3 & 1.1 & 2.4 \\\0.9 & -0.1 & 0.4 \\\\-0.7 & 0.3 & -0.5\end{array}\right] \quad B=\left[\begin{array}{cc}1.2 & -0.1 \\\0 & -0.5 \\\0.5 & -2.1\end{array}\right]$$ $$C=\left[\begin{array}{rrr}-0.2 & 0.2 & 0.1 \\\1.1 & 2.1 & -2.1\end{array}\right]$$ Use a graphing calculator to carry out the indicated algebraic operation, or explain why it cannot be performed. $$B+C$$

Step-by-Step Solution

Answer

The operation \( B+C \) cannot be performed due to incompatible matrix dimensions.

Step 1: Understand Dimensions of Matrices

Matrix \( B \) is a \( 3 \times 2 \) matrix, meaning it has 3 rows and 2 columns. Matrix \( C \) is a \( 2 \times 3 \) matrix, having 2 rows and 3 columns. To perform the addition of two matrices, both must have the same dimensions.

Step 2: Check Compatibility for Addition

Given that matrix \( B \) is \( 3 \times 2 \) and matrix \( C \) is \( 2 \times 3 \), they do not have the same dimensions. Hence, addition cannot be performed between \( B \) and \( C \).

Step 3: Conclusion on Operation

Since matrix addition requires matrices to be of the same size, and \( B \) and \( C \) do not match in dimensions, \( B + C \) is undefined.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. It involves combining two matrices by adding their corresponding elements. This process results in a new matrix of the same dimensions.
For example, given two matrices of dimensions \( m \times n \), adding them together means summing the elements at each position \((i, j)\). The resulting matrix will also have dimensions \( m \times n \).
However, for matrix addition to occur, both matrices must have exactly the same dimensions. This is essential because each element in one matrix is paired and added directly with the corresponding element in the other matrix.
If we were to take a matrix \( A \) of dimensions \( 2 \times 2 \) and another matrix \( D \) also of dimensions \( 2 \times 2 \), their addition \( A + D \) would see each element in \( A \) added to the corresponding element in \( D \).

• For example, if \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \) and \( D = \begin{bmatrix} d_{11} & d_{12} \ d_{21} & d_{22} \end{bmatrix} \), then \( A + D = \begin{bmatrix} a_{11} + d_{11} & a_{12} + d_{12} \ a_{21} + d_{21} & a_{22} + d_{22} \end{bmatrix} \).

Matrix Dimensions

Understanding the dimensions of matrices is crucial when performing operations like addition and multiplication.
Matrix dimensions are given as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.
For example, a matrix with 3 rows and 2 columns is represented as \( 3 \times 2 \). These dimensions provide insights into the matrix’s shape and size, allowing us to determine what operations can be performed between matrices.
Let's look at Matrix \( B \) and Matrix \( C \) from our problem:

• Matrix \( B = \begin{bmatrix} 1.2 & -0.1 \ 0 & -0.5 \ 0.5 & -2.1 \end{bmatrix} \) is \( 3 \times 2 \), having 3 rows and 2 columns.
• Matrix \( C = \begin{bmatrix} -0.2 & 0.2 & 0.1 \ 1.1 & 2.1 & -2.1 \end{bmatrix} \) is \( 2 \times 3 \), with 2 rows and 3 columns.

As such, these matrices differ in dimensions, meaning direct operations like addition are not straightforward. Knowing this helps avoid errors in calculations and ensures accuracy.
Always check the dimensions before attempting any computation, which will guide you on what is permissible or not.

Incompatible Matrices

When it comes to matrix operations, not all matrices can be combined through addition or other operations. Matrices that do not meet certain criteria for addition or multiplication are termed incompatible.

A common cause of matrix incompatibility is mismatched dimensions. As we've discussed, two matrices must share identical dimensions to be eligible for addition.
In our example:

• Matrix \( B \) is \( 3 \times 2 \)
• Matrix \( C \) is \( 2 \times 3 \)

Due to these differing dimensions, we identify \( B \) and \( C \) as incompatible for addition. Attempting to add such matrices results in undefined operations.
It's essential to understand that just because two matrices share the same number of elements doesn't mean they're compatible. It’s about the arrangement of these elements within the rows and columns that determine their compatibility.
Recognizing, understanding, and acknowledging matrix incompatibility is crucial in any mathematical context to prevent errors in computation. Always verify the dimensions to confirm compatibility before proceeding with operations.