Problem 1 - College Algebra with Corequisite Support

Question

Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

Step-by-Step Solution

Answer

Matrices must have the same dimensions to be added.

Step 1: Understanding Matrix Addition

The first step is to understand the fundamental rule of matrix addition. Matrices can only be added together if they have the same dimensions. For two matrices to have the same dimensions, they must have the same number of rows and the same number of columns.

Step 2: Explanation with an Example

Now, let's consider two matrices to illustrate this. Suppose we have: Matrix A, which is a 2x3 matrix: \[A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix} \]Matrix B, which is a 3x2 matrix: \[B = \begin{bmatrix} 7 & 8 \ 9 & 10 \ 11 & 12 \end{bmatrix} \].These two matrices cannot be added because they do not have the same dimensions. Matrix A has 2 rows and 3 columns, while Matrix B has 3 rows and 2 columns.

Step 3: Conclusion on Matrix Addition

From the previous step, it’s clear that not any two matrices can be added together. They must be of the same dimension. If matrices have different dimensions, like A and B in the example, the addition operation is undefined.

Key Concepts

Matrix Dimensions

When we talk about matrices in mathematics, one of the first things to understand is their dimensions. The dimensions of a matrix refer to how many rows and columns it has. For instance, if you see a matrix described as 2x3, this means it has 2 rows and 3 columns. It's like identifying a table's layout, where you count how many rows of data it has and how many columns each row contains.
Visualizing matrix dimensions is straightforward:

• The first number (2 in the 2x3 example) tells us how many horizontal lines of numbers there are, the rows.
• The second number (3 in the 2x3 example) indicates how many vertical lines of numbers fit into each row, the columns.

A good analogy is thinking of it as a book's pages. Each row is like a page, and each column is like a paragraph on that page. Understanding dimensions is crucial as it sets the stage for other matrix operations.

Matrices with the Same Dimensions

A fundamental rule in matrix addition is that only matrices with the same dimensions can be added together. If you picture matrices like grids, for two grids to be lined up perfectly, they need to be the same size.
Here’s what having the same dimensions means:

• Both matrices have identical numbers of rows.
• Both matrices have identical numbers of columns.

For example, a 2x2 matrix can only be added to another 2x2 matrix. This ensures that each element in one matrix has a corresponding element in the other matrix to add it to. Imagine trying to stack pieces of paper: sizes must match for them to align neatly. It’s the same principle with matrices. So, if you look at two matrices and see they don't have matching dimensions, you'll know immediately that they can't be added together.

Undefined Matrix Operations

An undefined matrix operation occurs when we try to perform a calculation that isn't allowed by the rules of matrix math. A common example is trying to add matrices that don't have the same dimensions. Imagine trying to put a square peg in a round hole—it just doesn’t fit.
Whenever you come across:

• Matrices with a different number of rows.
• Matrices with a different number of columns.

The addition operation is undefined, akin to a math error. Let's say you try to add a 2x3 matrix with a 3x2 matrix, like this:Matrix A = \[ \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix} \]Matrix B = \[ \begin{bmatrix} 7 & 8 \ 9 & 10 \ 11 & 12 \end{bmatrix} \]These simply won't combine through addition because their layouts don’t match. Each element in Matrix A lacks a direct counterpart in Matrix B to add with. The operation stops right there, marking itself as undefined.