Question

Matrices \(A\) and \(B\) are given below. Simplify the given expression. $$ A=\left[\begin{array}{cc} 1 & -1 \\ 7 & 4 \end{array}\right] \quad B=\left[\begin{array}{cc} -3 & 2 \\ 5 & 9 \end{array}\right] $$ $$ A+B $$

Short answer

\(A+B = \begin{pmatrix} -2 & 1 \\ 12 & 13 \end{pmatrix}\).

Step-by-step solution

Identify Matrix Elements

Write down the elements of matrix \(A\) and matrix \(B\). Matrix \(A\) has elements \(a_{11}=1\), \(a_{12}=-1\), \(a_{21}=7\), \(a_{22}=4\), and matrix \(B\) has elements \(b_{11}=-3\), \(b_{12}=2\), \(b_{21}=5\), \(b_{22}=9\).

Addition of Corresponding Elements

Add the corresponding elements of matrices \(A\) and \(B\). That is, calculate each element of the resultant matrix \(C = A+B\): \(c_{11} = a_{11} + b_{11} = 1 + (-3)\), \(c_{12} = a_{12} + b_{12} = -1 + 2\), \(c_{21} = a_{21} + b_{21} = 7 + 5\), \(c_{22} = a_{22} + b_{22} = 4 + 9\).

Simplify Each Expression

Perform the arithmetic for each calculated element: \(c_{11} = 1 + (-3) = -2\), \(c_{12} = -1 + 2 = 1\), \(c_{21} = 7 + 5 = 12\), \(c_{22} = 4 + 9 = 13\).

Write the Resultant Matrix

Form the resulting matrix \(C\) using the simplified elements: \[C = \begin{pmatrix} -2 & 1 \ 12 & 13 \end{pmatrix}\].

Key concepts

Element-wise Operations

In matrix addition, the term **element-wise operations** refers to the process of adding corresponding elements from two matrices. Essentially, for two matrices of the same dimensions, each element of the first matrix is paired with the element in the same position of the second matrix. This is a fundamental operation in matrix arithmetic, ensuring that each individual cell in the matrix is addressed.

Consider matrices with dimensions 2x2, denoted as matrix \( A \) and matrix \( B \). For matrix addition to be possible, both matrices must have identical dimensions. The elements are summed simply as follows:
\( c_{ij} = a_{ij} + b_{ij} \).

This means:

• Take the element at row 1, column 1 of matrix \( A \) and add it to the element at row 1, column 1 of matrix \( B \) to get \( c_{11} \).
• Repeat this process for other elements: \( c_{12} \), \( c_{21} \), and \( c_{22} \).

Addition is performed element by element, always ensuring that you are combining elements positioned at the same index in each matrix.

Matrix Arithmetic

**Matrix arithmetic** involves various operations like addition, subtraction, and multiplication of matrices. In this context, we focus on matrix addition. It's crucial to understand its rules and the conditions that must be satisfied for these operations to be valid.

• Dimensions Requirement: For two matrices to be added, they must possess the same dimensions. This means a matrix with rows \( m \) and columns \( n \) can only be added to another matrix with \( m \) rows and \( n \) columns.
• Addition Operation: The addition process is executed by summing elements that are in the same position. Think of it like overlapping two matrices of the same shape and adding each stacked layer together.

The result of adding two matrices is a new matrix where each element is the sum of the corresponding elements of the original matrices.
For the matrices \( A \) and \( B \) given above, the matrix arithmetic operation led to matrix \( C \), calculated as:
\[ C = \begin{pmatrix} -2 & 1 \ 12 & 13 \end{pmatrix} \].

Linear Algebra Concepts

**Linear algebra concepts** provide the foundation for many operations, including matrix addition. It's a branch of mathematics dealing with vectors, matrices, and linear transformations.
Understanding matrix operations is vital in solving system equations and analyzing linear transformations.

Some key linear algebra concepts include:

• Matrices: Rectangular arrays of numbers or functions arranged in rows and columns.
• Vector Spaces: A set of vectors that can be added together or multiplied by scalars.
• Linear Mapping: Refers to functions that respect the operations of addition and scalar multiplication.

Matrix addition is one of the first operations students often encounter in linear algebra. It simplifies systems of equations, allowing for the representation of complex linear transformations in a manageable format.
Mastering these concepts opens doors to more advanced topics within mathematics and its applications to science and engineering.