Problem 37 - Algebra for College Students

Question

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \((A+B)+C=A+(B+C)\).

Step-by-Step Solution

Answer

The associative property of addition proves that \((A+B)+C = A+(B+C)\).

Step 1: Understand the Matrix Addition

Matrix addition involves adding corresponding elements in the matrices. For any two matrices \(M\) and \(N\), \((M+N)_{ij} = M_{ij} + N_{ij}\), where \((i,j)\) indicates the position of the element.

Step 2: Calculate \((A+B)\)

The matrix \(A+B\) is obtained by adding corresponding elements of matrices \(A\) and \(B\):\[A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}\]

Step 3: Calculate \((A+B)+C\)

Now, add matrix \(C\) to \(A+B\) by adding corresponding elements:\[(A+B) + C = \begin{bmatrix} (a_{11} + b_{11}) + c_{11} & (a_{12} + b_{12}) + c_{12} \ (a_{21} + b_{21}) + c_{21} & (a_{22} + b_{22}) + c_{22} \end{bmatrix}\]

Step 4: Calculate \((B+C)\)

The matrix \(B+C\) is obtained by adding corresponding elements of matrices \(B\) and \(C\):\[B + C = \begin{bmatrix} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{bmatrix}\]

Step 5: Calculate \(A+(B+C)\)

Now, add matrix \(A\) to \(B+C\) by adding corresponding elements:\[A + (B+C) = \begin{bmatrix} a_{11} + (b_{11} + c_{11}) & a_{12} + (b_{12} + c_{12}) \ a_{21} + (b_{21} + c_{21}) & a_{22} + (b_{22} + c_{22}) \end{bmatrix}\]

Step 6: Compare \((A+B)+C\) and \(A+(B+C)\)

Since the addition of elements is associative, the results for both \((A+B) + C\) and \(A + (B+C)\) are the same:\[\begin{bmatrix} (a_{11} + b_{11}) + c_{11} & (a_{12} + b_{12}) + c_{12} \ (a_{21} + b_{21}) + c_{21} & (a_{22} + b_{22}) + c_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + (b_{11} + c_{11}) & a_{12} + (b_{12} + c_{12}) \ a_{21} + (b_{21} + c_{21}) & a_{22} + (b_{22} + c_{22}) \end{bmatrix}\]

Step 7: Conclusion

The matrix expression \((A+B)+C = A+(B+C)\) holds true due to the associative property of matrix addition.

Key Concepts

Associative Property

The associative property is a fundamental rule in both mathematics and matrix algebra. It tells us that when adding matrices, the grouping of the matrices does not affect the sum. In simple terms, if you have three matrices, say, \( A \), \( B \), and \( C \), you can add them together in any sequence and the result will be the same.
For example:

• \((A + B) + C = A + (B + C)\)

This principle applies to numbers and to matrices alike. When we deal with matrices, this property ensures flexibility in the order of operations. This flexibility can be particularly helpful when dealing with complex calculations, allowing us to simplify parts of matrix algebra without changing the end result.

Matrix Algebra

Matrix algebra is a field of mathematics focused on the study and manipulation of matrices. Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. Matrix operations, such as matrix addition, subtraction, multiplication, and finding determinants, are fundamental applications in various fields like computer science, physics, and statistics.
When adding matrices, a key point is that they must be of the same size—meaning the number of rows and columns must match. For each element in the resulting matrix, you take the corresponding elements from each of the matrices being added and sum them together. It's much like adding numbers, but done in a grid format.

• Matrix addition is denoted as \((M+N)_{ij} = M_{ij} + N_{ij}\)

Learning how to add matrices is essential in understanding more complex operations like matrix multiplication, solving linear equations, and performing transformations in geometry.

Linear Algebra Concepts

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, and is fundamental to understanding systems of linear equations, vector spaces, determinants, eigenvalues, and matrices.
One of the basic operations in linear algebra is matrix addition, which is essential for solving systems of linear equations. Matrix representation simplifies computations and provides a visual way to handle linear transformations.

• Matrices can represent coefficients in linear equations.
• Matrix operations are used to perform transformations and solve systems efficiently.

The associative property of matrix addition is one of the many properties that help in manipulating matrices and simplifying expressions in linear algebra. This foundation is critical for more advanced applications, such as transformation matrices in computer graphics, algorithms in statistics, and models in machine learning.