Problem 11 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand Matrix Dimensions

To determine whether the subtraction \(D - B\) is possible, we need to check the dimensions of matrices \(D\) and \(B\). Matrix \(D\) is a 3x2 matrix, and matrix \(B\) is a 2x2 matrix. For subtraction to be defined, both matrices must have the same dimensions.

Step 2: Determine the Operation Possibility

Since matrix \(D\) has dimensions 3x2 and matrix \(B\) has dimensions 2x2, these matrices cannot be directly subtracted from each other because they do not have the same dimensions.

Step 3: Conclude on Matrix Subtraction

The subtraction \(D - B\) is undefined because the dimensions of the matrices do not match. Only matrices with the same number of rows and columns can be subtracted or added together.

Key Concepts

matrix addition

Matrix addition is a fundamental operation in linear algebra. It involves adding two matrices together to produce a new matrix.
To add two matrices, they must have the same dimensions. This means both matrices need to have the same number of rows and columns.
Here's how matrix addition works step-by-step:

• Ensure both matrices have identical dimensions (e.g., both are 2x2).
• Add corresponding elements from each matrix.
• The resulting matrix will have elements where each piece is the sum of the corresponding elements from the original matrices.

For example, if we add matrix A \[A=\begin{bmatrix}1 & 3 \ 0 & 7\end{bmatrix}\] with matrix E \[E=\begin{bmatrix}6 & 12 \ 14 & 5\end{bmatrix}\], the resulting matrix would be found by adding each corresponding element: \[A+E=\begin{bmatrix}1+6 & 3+12 \ 0+14 & 7+5\end{bmatrix}=\begin{bmatrix}7 & 15 \ 14 & 12\end{bmatrix}\].
Matrix addition is straightforward when you remember to align your rows and columns!

matrix subtraction

Matrix subtraction is quite similar to matrix addition, but instead of adding corresponding elements, you subtract them. This operation is only possible when the matrices involved have the same dimensions.
Here's how:

• First, check that both matrices have the same dimensions.
• Subtract each element in the second matrix from the corresponding element in the first matrix.
• The result is a new matrix with each element showing the difference between the corresponding elements of the original matrices.

For example, the matrices A and E can be subtracted like this:\[A-E=\begin{bmatrix}1-6 & 3-12 \ 0-14 & 7-5\end{bmatrix}=\begin{bmatrix}-5 & -9 \ -14 & 2\end{bmatrix}\].
Remember, the matrices must be of the same size for subtraction to proceed!

matrix dimensions

Matrix dimensions refer to the size of the matrix, described in terms of the number of rows and columns it contains. Understanding dimensions is crucial when performing matrix operations like addition and subtraction.
The dimension of a matrix is written as rows x columns. For example, a 3x2 matrix has 3 rows and 2 columns.
It's important because:

• For matrix addition and subtraction, the matrices must have the same dimensions.
• Different operations require matrices of specific dimensions, such as matrix multiplication, which has its own rules.
• Operations are undefined if dimensions don't match, which is crucial to identify in problems.

In the previous scenario, matrices D and B could not be subtracted because D is a 3x2 matrix while B is 2x2. The mismatch in rows makes subtraction impossible.

undefined operations

Undefined operations in matrices occur when the necessary conditions for the operation are not met, primarily involving dimension mismatches.
Here's why an operation would be undefined:

• Matrix addition or subtraction requires matrices to have the same dimensions.
• If a matrix operation is attempted on matrices with differing row or column counts, the operation cannot be completed.

For instance, subtracting matrix B from matrix D is undefined because D is 3x2, and B is 2x2.
This dimension mismatch is a common reason operations remain undefined. A good practice is to always check dimensions before attempting any matrix operations to avoid these undefined cases.