Problem 19 - Algebra and Trigonometry

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as follows. $$A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right]$$ $$\begin{array}{l}{D=\left[\begin{array}{lll}{7} & {3}\end{array}\right]} & {E=\left[\begin{array}{lll}{1} \\ {1} \\ {2} \\ {0}\end{array}\right]} \\\ {F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] \quad G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right]}\end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ C-B $$

Step-by-Step Solution

Answer

\(C - B = \begin{bmatrix} -1 & -3 & -5 \\ -1 & 3 & -6 \end{bmatrix}\)

Step 1: Confirm Compatibility

Before performing matrix subtraction, ensure both matrices have the same dimensions. Matrix \(C\) is a \(2 \times 3\) matrix and Matrix \(B\) is also a \(2 \times 3\) matrix. Therefore, they are compatible for subtraction.

Step 2: Subtract the Corresponding Elements

To subtract two matrices, subtract corresponding elements. If \(C = [c_{ij}]\) and \(B = [b_{ij}]\), then \(C - B = [c_{ij} - b_{ij}]\). Compute each element:- First row, first column: \(2 - 3 = -1\)- First row, second column: \(-\frac{5}{2} - \frac{1}{2} = -3\)- First row, third column: \(0 - 5 = -5\)- Second row, first column: \(0 - 1 = -1\)- Second row, second column: \(2 - (-1) = 3\)- Second row, third column: \(-3 - 3 = -6\)

Step 3: Write the Resultant Matrix

The result of subtracting matrix \(B\) from \(C\) is the matrix:\[C - B = \left[\begin{array}{rrr}-1 & -3 & -5 \-1 & 3 & -6\end{array}\right]\]

Key Concepts

Matrix Compatibility

Matrix subtraction is an operation that requires the two matrices involved to have the same dimensions. This means both matrices need to have the same number of rows and the same number of columns for their subtraction to be feasible.
When we refer to two matrices being compatible for subtraction, we are solely concerned with their dimensions being the same.

• For example, a 2x3 matrix can only be subtracted from another 2x3 matrix.
• This ensures that every corresponding element from one matrix has a matching element in the other matrix to subtract from.

In the given exercise, matrix \(C\) and matrix \(B\) are both 2x3 matrices. Hence, they are compatible for the subtraction process.

Matrix Dimensions

Matrix dimensions are often indicated in terms of "rows by columns," meaning you first count the number of rows then the number of columns. This characteristic is crucial in determining the compatibility of two matrices for addition or subtraction.
Understanding these dimensions helps in predicting the shape of the result after performing operations like subtraction.

• For instance, a matrix described as 3x4 has 3 rows and 4 columns.
• The dimensions tell you not only about its size but also about the potential compatibility with other matrices.

In the problem, matrix \(C\) and matrix \(B\) both have dimensions of 2x3, allowing us to proceed with subtraction.

Element-Wise Operations

Element-wise operations mean handling individual elements separately when performing matrix arithmetic. In the case of subtraction, each element in one matrix is subtracted from the corresponding element in the other matrix.
The operation needs to respect the position of elements, ensuring that only identical elements (same row, same column) from two matrices are paired in the subtraction.

• For example, if you have elements \(c_{11}\) and \(b_{11}\), the subtraction yields \(c_{11} - b_{11}\).
• This is applied throughout the matrices, row by row and column by column.

In the step-by-step solution provided, elements of matrix \(C\) are subtracted from those of matrix \(B\) element by element. This results in the new matrix where each position reflects the subtraction of the original pairs, such as \(2 - 3 = -1\) for the first elements of the first row.