Problem 11 - College Algebra

Question

Use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 7\end{array}\right], B=\left[\begin{array}{rr}2 & 14 \\ 22 & 6\end{array}\right], C=\left[\begin{array}{rr}1 & 5 \\ 8 & 92 \\ 12 & 6\end{array}\right], D=\left[\begin{array}{rr}10 & 14 \\ 7 & 2 \\ 5 & 61\end{array}\right], E=\left[\begin{array}{rr}6 & 12 \\ 14 & 5\end{array}\right], F=\left[\begin{array}{rr}0 & 9 \\ 78 & 17 \\ 15 & 4\end{array}\right]\) \(D-B\)

Step-by-Step Solution

Answer

The operation \( D - B \) is undefined due to mismatched matrix dimensions.

Step 1: Check matrix dimensions

First, we check the dimensions of matrices \( D \) and \( B \). Matrix \( D \) is a \(3 \times 2\) matrix, and matrix \( B \) is a \(2 \times 2\) matrix. For matrix subtraction to occur, the matrices need to have identical dimensions.

Step 2: Determine if subtraction is defined

Since matrix \( D \) has dimensions \(3 \times 2\) and matrix \( B \) has dimensions \(2 \times 2\), they do not match. Therefore, the subtraction \( D - B \) is undefined.

Key Concepts

Matrix Dimensions

One of the first and most crucial steps in performing any matrix operation is to check the dimensions of the involved matrices. Understanding matrix dimensions is like knowing the size of puzzle pieces before trying to fit them together. Each matrix has a certain size known as its dimensions, described by the number of rows and columns it has. For instance:

• A \(3 \times 2\) matrix means it has 3 rows and 2 columns.
• A \(2 \times 2\) matrix has 2 rows and 2 columns, making it a square matrix.

To perform operations like addition or subtraction, matrices must share the same dimensions:

• Count the number of rows in both matrices.
• Count the number of columns in both matrices.
• If both are identical, you can proceed with your operation!

However, if the dimensions between two matrices differ, addition or subtraction is not possible. Just like trying to fit two incompatible puzzle pieces together won't work, attempting to operate on matrices with different sizes will be futile.

Undefined Operations

Matrix operations such as addition or subtraction are described as undefined when the matrices involved do not have matching dimensions. This scenario occurs when it's impossible to align each element from one matrix with a corresponding element in the other matrix.
In our specific situation with matrices \(D\) and \(B\), we found out that:

• Matrix \(D\) is a \(3 \times 2\) matrix.
• Matrix \(B\) is a \(2 \times 2\) matrix.

The number of rows and columns between the two matrices does not match. Consequently, any attempt to subtract matrix \(B\) from \(D\) is undefined.
Practically, it means there's no corresponding element in each position of those two matrices, which makes the operation invalid. Always remember, matching dimensions are essential for defined and successful matrix additions and subtractions. If they don’t align, the operation simply cannot be carried out.

Matrix Addition

Matrix addition is a straightforward process that resembles ordinary addition but is done on a much larger scale. This operation involves adding every corresponding element of two matrices together. In order to perform matrix addition, the matrices need to be of identical dimensions. This similarities ensure that each position in one matrix has a counterpart in the other matrix.
For example: suppose we have two matrices:

• \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \)
• \( \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} \)

If we add both, we simply add the corresponding elements together:

• \( 1 + 5 = 6 \) in position (1,1)
• \( 2 + 6 = 8 \) in position (1,2)
• \( 3 + 7 = 10 \) in position (2,1)
• \( 4 + 8 = 12 \) in position (2,2)

This results in a new matrix:
\( \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix} \)
Pay attention to always check the dimensions first. If they don't match, the addition is not possible!

Matrix Subtraction

Matrix subtraction works similarly to matrix addition but involves subtracting corresponding elements of one matrix from another. It's like reversing addition but only applicable when matrices share identical dimensions.
This ensures that for every element in the first matrix, there's a corresponding element in the second matrix to subtract from.

• Example:\(\begin{bmatrix} 8 & 12 \ 14 & 5 \end{bmatrix}\) and\(\begin{bmatrix} 6 & 10 \ 4 & 7 \end{bmatrix}\)
• Subtract each element:\(8 - 6 = 2\) in position (1,1)
• \(12 - 10 = 2\) in position (1,2)
• \(14 - 4 = 10\) in position (2,1)
• \(5 - 7 = -2\) in position (2,2)

Evidently, the resulting matrix is:\(\begin{bmatrix} 2 & 2 \ 10 & -2 \end{bmatrix}\)
Always verify that the dimensions align. If they differ, just like with addition, the subtraction is not defined.