Problem 16 - Algebra 2

Question

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{ccc}{-5} & {2} & {-1}\end{array}\right]+\left[\begin{array}{r}{2} \\ {-2} \\\ {1}\end{array}\right] $$

Step-by-Step Solution

Answer

Impossible to add due to different dimensions.

Step 1: Analyze the Dimensions

Identify the dimensions of the given matrices. The first matrix is \([-5\ 2\ -1]\), which is a 1x3 matrix. The second matrix is \(\begin{bmatrix} 2 \ -2 \ 1 \end{bmatrix}\), which is a 3x1 matrix. These matrices have different dimensions, hence we need to check if they can be added.

Step 2: Determine If Addition is Possible

Matrix addition is only possible if the matrices have the same dimensions. Here, the first matrix is 1x3 and the second matrix is 3x1. Since their dimensions do not match, the matrices cannot be added.

Step 3: Conclusion

Since the matrices have different dimensions (1x3 and 3x1), it is impossible to perform the addition operation.

Key Concepts

matrix dimensions

When dealing with matrices, understanding their dimensions is essential. A matrix's dimensions are defined by the number of rows and columns it contains. Each matrix is expressed in the form of "rows x columns."
For instance, a matrix with 3 rows and 2 columns is called a 3x2 matrix. These dimensions directly affect the operations you can perform on the matrix:

• In a 1x3 matrix, there is 1 row and 3 columns.
• In a 3x1 matrix, there are 3 rows and 1 column.

Correctly understanding and identifying matrix dimensions allows you to determine if various matrix operations, like addition or multiplication, are possible.

impossible matrix operations

Matrix operations are not always straightforward, especially when it comes to addition and multiplication. For addition and subtraction, matrices must have identical dimensions. This means the number of rows and columns must match.

• An operation becomes impossible if these dimensions do not match, because each element in one matrix must have a corresponding element in the other matrix.
• This is why a 1x3 matrix and a 3x1 matrix cannot be added. They have different numbers of rows and columns, making addition impossible.

Recognizing when matrix operations are impossible is a key component of working with matrices in mathematical problems.

1x3 matrix

A 1x3 matrix has only one row and three columns. It looks like a single-row table and can be written horizontally.
For example, the matrix \([-5\ 2\ -1]\) is a typical 1x3 matrix. You might encounter such matrices in various applications, like representing variables or coefficients in a linear equation.

• Each element in the matrix is separated by spaces or commas.
• This format is straightforward for representing simple data sets.

Because of its structure, it cannot be directly added to matrices with different dimensions.

3x1 matrix

A 3x1 matrix is comprised of 3 rows and a single column. It is often referred to as a column matrix and is written vertically.
For instance, \(\begin{bmatrix} 2 \ -2 \ 1 \end{bmatrix}\) is a 3x1 matrix. Such matrices frequently appear in contexts like vectors or statistics where data is naturally aligned in a column.

• Each entry is written below the previous one, creating a vertical list.
• This makes it useful for representing sets of linear transformations or projections.

However, due to its singular column structure, it cannot be added to matrices with different dimensions.