Question

Find the sum of the matrices. \(\left[\begin{array}{rc}4 & 10 \\ -1 & 9\end{array}\right]+\left[\begin{array}{rr}-20 & 40 \\ -8 & 10\end{array}\right]\)

Short answer

The sum of the matrices is \(\left[\begin{array}{rc}-16 & 50 \ -9 & 19\end{array}\right]\)

Step-by-step solution

Identify the corresponding elements

It's important to know that matrices can only be added if they have the same dimensions. In this case, both matrices are 2x2, so they can be added. The corresponding elements are the elements in the same position in each matrix. That is, the top left elements of both matrices, the top right elements of both matrices, the bottom left elements of both matrices, and the bottom right elements of both matrices.

Add the corresponding elements

The sum of the matrices is found by adding the corresponding elements from each matrix together. The top left element of the sum is \(4+(-20)\), the top right element is \(10+40\), the bottom left element is \(-1+(-8)\) and the bottom right element is \(9+10\).

Calculate the sums

Adding those together gives us the matrix \(\left[\begin{array}{rc}-16 & 50 \ -9 &19\end{array}\right]\)

Key concepts

Matrices Operations

Matrix operations, including addition, subtraction, and multiplication, are fundamental concepts in linear algebra. When we talk about adding two matrices, the crucial condition is that they must be of the same size. They could be 2x2 matrices, 3x3, or any other size, as long as both matrices have the same number of rows and columns.

During matrix addition, we calculate the sum by taking each element from one matrix and adding it to the corresponding element in the other matrix. It's almost like combining two identical grids of numbers. This operation can only be done with matrices of the same dimensions because the concept of a corresponding element does not exist if the matrices differ in size.

For students to tackle these problems efficiently, it's helpful to visualize the operation as merging two layered grids and calculating the new values for each position. Imagine laying two transparent sheets, each with a grid of numbers, on top of each other and summing the numbers that align. Additionally, using colored pens or highlighting the corresponding elements can be a good visual aid for students who may be struggling to keep track of the positions during addition.

Corresponding Elements

The term 'corresponding elements' is used in matrix operations to refer to elements in the same position within different matrices. For example, in a 2x2 matrix, the element in the first row and first column of the first matrix corresponds to the element in the first row and first column of the second matrix. They are like matching puzzle pieces.

Understanding this concept is crucial when adding matrices. The rule of thumb is: only the corresponding elements can be combined with each other. This means you take the numbers that occupy the same 'address' within each matrix and perform the addition.

Visual Aid for Corresponding Elements

A technique that can be quite effective for visual learners is to physically mark corresponding elements with symbols or colors. This way, students can quickly identify which elements to add together, reducing the likelihood of mixing up positions during calculations.

It's also noteworthy to mention that if a student is ever in doubt about which elements are corresponding, they can refer to the row and column indices of the elements. The indices act as a geographical 'coordinate system' for pinpointing the exact location of an element within the matrix landscape.

2x2 Matrices

2x2 matrices are the simplest form of square matrices and are particularly important for building the fundamental skills needed for bigger and more complex matrix operations. They consist of four elements arranged into two rows and two columns. Understanding operations on 2x2 matrices sets the precedent for understanding larger matrices.

Due to their small size, 2x2 matrices provide an excellent starting point for practicing the concept of corresponding elements and matrix addition. They are easy to visualize and the calculations are straightforward, which is beneficial for beginners.

Practice Tips for 2x2 Matrices

Students can gain confidence by manually adding several 2x2 matrices before moving onto larger sized matrices. When practicing with 2x2 matrices, it's useful to recognize patterns in the addition process. For instance, the 'diagonal' elements (top left and bottom right) often carry certain properties in advanced operations, such as determinants and inverses, thus getting familiar with such patterns early on is advantageous.