Problem 9 - 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. \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 7\end{array}\right], B=\left[\begin{array}{cc}2 & 14 \\ 22 & 6\end{array}\right], C=\left[\begin{array}{cc}1 & 5 \\ 8 & 92 \\ 12 & 6\end{array}\right], D=\left[\begin{array}{cc}10 & 14 \\ 7 & 2 \\ 5 & 61\end{array}\right], E=\left[\begin{array}{cc}6 & 12 \\ 14 & 5\end{array}\right], F=\left[\begin{array}{cc}0 & 9 \\ 78 & 17 \\ 15 & 4\end{array}\right]\) $$ B-E $$

Step-by-Step Solution

Answer

Subtract corresponding elements: \(B - E = \begin{bmatrix} -4 & 2 \\ 8 & 1 \end{bmatrix}\).

Step 1: Determine Matrix Dimensions

First, identify the dimensions of matrices \(B\) and \(E\). Matrix \(B\) is a 2x2 matrix, and matrix \(E\) is also a 2x2 matrix.

Step 2: Check Compatibility for Subtraction

Matrix subtraction is defined when the two matrices have the same dimensions. Since both \(B\) and \(E\) are 2x2 matrices, matrix subtraction is possible.

Step 3: Perform Subtraction Element-wise

Subtract the corresponding elements of matrix \(E\) from matrix \(B\):\[B - E = \begin{bmatrix} 2 - 6 & 14 - 12 \ 22 - 14 & 6 - 5 \end{bmatrix} = \begin{bmatrix} -4 & 2 \ 8 & 1 \end{bmatrix}\]

Step 4: Verify Results

Check each element subtraction to ensure all operations were correctly performed:- Top left: \(2 - 6 = -4\)- Top right: \(14 - 12 = 2\)- Bottom left: \(22 - 14 = 8\)- Bottom right: \(6 - 5 = 1\)The results match the calculated matrix, confirming the subtraction is correct.

Key Concepts

Matrix Addition

Matrix addition is a fundamental concept in linear algebra. It involves adding two matrices by performing addition on their corresponding elements. To do this, each element in one matrix is added to the element in the same position in the other matrix.
For example, if you have matrices \(A\) and \(B\):

• Matrix \(A = \begin{bmatrix} 1 & 3 \ 0 & 7 \end{bmatrix}\)
• Matrix \(B = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}\)

To add these, you would sum each corresponding element:

• Top left: \(1 + 2 = 3\)
• Top right: \(3 + 4 = 7\)
• Bottom left: \(0 + 6 = 6\)
• Bottom right: \(7 + 8 = 15\)

The result of the addition is a new matrix \(C = \begin{bmatrix} 3 & 7 \ 6 & 15 \end{bmatrix}\). It is important to note that matrix addition requires matrices to have the same dimensions.
If they don't, the operation is undefined.

Matrix Subtraction

Matrix subtraction is quite similar to matrix addition but involves subtracting elements instead. This operation is only possible when the matrices involved share the same dimensions.
Take matrices \(B\) and \(E\) as an example:

• Matrix \(B = \begin{bmatrix} 2 & 14 \ 22 & 6 \end{bmatrix}\)
• Matrix \(E = \begin{bmatrix} 6 & 12 \ 14 & 5 \end{bmatrix}\)

To subtract matrix \(E\) from matrix \(B\), you subtract corresponding elements:

• Top left: \(2 - 6 = -4\)
• Top right: \(14 - 12 = 2\)
• Bottom left: \(22 - 14 = 8\)
• Bottom right: \(6 - 5 = 1\)

The result is a new matrix \(F = \begin{bmatrix} -4 & 2 \ 8 & 1 \end{bmatrix}\).
Always ensure the dimensions match, as matrix subtraction is undefined otherwise.

Matrix Dimensions

Understanding matrix dimensions is crucial when performing matrix operations. The dimensions are described by "rows x columns." This simple format helps quickly identify the size of the matrix.
Let's consider matrices \(B\) and \(E\):

• Matrix \(B = \begin{bmatrix} 2 & 14 \ 22 & 6 \end{bmatrix}\) is described as 2x2 because it has 2 rows and 2 columns.
• Matrix \(E = \begin{bmatrix} 6 & 12 \ 14 & 5 \end{bmatrix}\) is also a 2x2 matrix.

The dimensions dictate whether operations such as addition or subtraction can occur.
It's essential because matrices must have matching dimensions for these element-wise operations to be defined and performed correctly.

• If you have a 2x2 matrix, it can only perform operations with another 2x2 matrix.

Element-wise Operations

Element-wise operations refer to processing matrices by addressing each corresponding element. In this context, operations like addition and subtraction occur one element at a time.
For each position, whether it's the upper left or lower right, the elements are taken from the same indices in each matrix and then operated on like simple numbers.
Consider matrices \(B\) and \(E\) once more:

• Matrix \(B = \begin{bmatrix} 2 & 14 \ 22 & 6 \end{bmatrix}\)
• Matrix \(E = \begin{bmatrix} 6 & 12 \ 14 & 5 \end{bmatrix}\)

To subtract these matrices, you perform simple subtraction for each corresponding element.
Element-wise operations ensure that every number in a matrix is treated with precision, allowing accurate and defined results.
Remember, these operations require the matrices to have the same dimensions, ensuring each element has a counterpart to be operated on.