Problem 11 - College Algebra

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 because their dimensions do not match.

Step 1: Determine Matrix Dimensions

First, 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.

Step 2: Verify Compatibility for Subtraction

To subtract two matrices, their dimensions must be identical. Since \( D \) is \( 3 \times 2 \) and \( B \) is \( 2 \times 2 \), their dimensions are not the same. This makes the operation undefined.

Step 3: Conclude the Operation

Since the dimensions of matrices \( D \) and \( B \) are different, the subtraction \( D - B \) is undefined. No further calculation is possible in this case.

Key Concepts

Matrix Addition

Matrix addition is when you combine two matrices by adding their corresponding elements element-by-element. It's quite similar to how you might add numbers in a coordinate grid. This operation is only possible when the matrices involved have the exact same dimensions. For instance, if you have two matrices, both of size \(2 \times 2\), you can add them easily.

• Add the first element from the first row and first column of the first matrix to the first element from the first row and first column of the second matrix.
• Repeat this process for each corresponding element.

This results in a new matrix of the same dimensions as the original two. Remember: if the dimensions aren't the same, you cannot perform the operation as it is undefined.

Matrix Subtraction

Matrix subtraction works much like matrix addition. The main difference is that instead of adding the corresponding elements, you subtract them. For this operation, the two matrices must also have the same dimensions.

• Subtract the first element of the first matrix from the first element of the second matrix.
• Continue subtracting corresponding elements until you finish the matrix.

Matrix subtraction is undefined if the dimensions of the two matrices are unequal—this is crucial. Always ensure the dimensions match before you begin any subtraction.

Matrix Dimensions

The dimensions of a matrix describe its size in terms of its rows and columns. It is often expressed as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.

• A matrix with two rows and three columns is a \(2 \times 3\) matrix.
• Matrices can only be added or subtracted if their dimensions are identical.

Checking dimensions is the first and most critical step in performing any matrix operation. If they don’t match, then these operations cannot proceed. Always check dimensions first before trying to compute further.

Undefined Operations

In matrix operations, undefined operations frequently occur when trying to perform actions like addition or subtraction with matrices that don't share the same dimensions.

• If you attempt to add a \(3 \times 3\) matrix with a \(2 \times 3\) matrix, the result is undefined.
• Also, if you try to subtract matrices of different sizes, the operation can't be performed and is labeled as undefined.

The concept of undefined operations reminds us to be vigilant about dimensions first. This situation helps in identifying steps to correct erroneous matrix manipulations by understanding the fundamental requirement of equal dimensions.