Problem 9 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result of \( B - E \) is \( \left[ \begin{array}{cc} -4 & 2 \\ 8 & 1 \end{array} \right] \).

Step 1: Verify Matrix Dimensions

Before we can add or subtract matrices, we need to ensure they are of the same dimensions. Matrix \( B \) is a \( 2 \times 2 \) matrix and matrix \( E \) is also a \( 2 \times 2 \) matrix.

Step 2: Perform Element-wise Subtraction

Since both \( B \) and \( E \) are \( 2 \times 2 \) matrices, we can subtract them by performing element-wise subtraction. The resulting matrix \( B - E \) is:\[\left[ \begin{array}{cc} 2-6 & 14-12 \ 22-14 & 6-5 \end{array} \right] = \left[ \begin{array}{cc} -4 & 2 \ 8 & 1 \end{array} \right] \]

Key Concepts

Matrix Addition

Matrix addition is a simple operation that involves adding corresponding elements of two matrices. The key requirement for matrix addition is that both matrices should have the same dimensions.

For instance, if we have two matrices, say, matrix \( A \) and matrix \( B \), their dimensions must match for them to be added. This means they should have the same number of rows and columns.

• If matrix \( A \) is of dimension \( m \times n \), then matrix \( B \) must also be \( m \times n \).

This ensures every element in matrix \( A \) has a corresponding element in matrix \( B \) to add.

After verifying the matrices have the same dimensions, you simply add each element from matrix \( A \) to the matching element in matrix \( B \). For example: \[ \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] + \left[ \begin{array}{cc} e & f \ g & h \end{array} \right] = \left[ \begin{array}{cc} a+e & b+f \ c+g & d+h \end{array} \right] \]
This operation is straightforward and follows element-wise addition.

Matrix Subtraction

Matrix subtraction is similar to matrix addition but involves subtracting elements instead. The key rule here is again that both matrices must have the same dimensions.

Like in the exercise provided, let's consider matrices \( B \) and \( E \) as an example. Both matrices must have exactly the same number of rows and columns to perform subtraction.

• For example, if matrix \( B \) has dimensions \( 2 \times 2 \), the same must be true for matrix \( E \).

To perform matrix subtraction, subtract each element in matrix \( E \) from the corresponding element in matrix \( B \).

This process looks like: \[ \left[ \begin{array}{cc} b & f \ g & j \end{array} \right] - \left[ \begin{array}{cc} e & d \ h & k \end{array} \right] = \left[ \begin{array}{cc} b-e & f-d \ g-h & j-k \end{array} \right] \]
This element-wise subtraction results in a new matrix that is the same size as the original matrices.

Matrix Dimensions

Understanding matrix dimensions is crucial when performing certain matrix operations, such as addition and subtraction. The dimensions of a matrix are expressed as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.

When working with matrices, dimensions determine whether operations like addition or subtraction can be performed. For example, in the given exercise, both matrices \( B \) and \( E \) have dimensions \( 2 \times 2 \).

• This means they each have 2 rows and 2 columns.
• For the operation to be valid, the matrices must share these dimensions.

Dimension verification acts like a compatibility check before performing element-wise operations.

Always check dimensions first to avoid undefined operations. A mismatch in dimensions would mean matrix operations such as addition or subtraction can't be executed. For instance, a matrix \( C \) with dimensions \( 3 \times 2 \) cannot be added to or subtracted from a \( 2 \times 2 \) matrix.