Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{ll}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(C-0.5 D\)

Step-by-Step Solution

Answer

The operation \( C - 0.5D \) cannot be performed due to different matrix dimensions.

Step 1: Understand the matrices dimensions

Matrix \( C \) is a 2x2 matrix, and matrix \( D \) is a 3x3 matrix. Since matrix subtraction requires both matrices to be of the same dimensions, it's crucial to check whether the operation \( C - 0.5D \) can be performed. Due to the dimension mismatch, the operation is not immediately obvious.

Step 2: Check for operation feasibility

Check whether the operation can be performed considering the matrices involved. Since \( C \) is 2x2 and \( D \) is 3x3, these matrices do not conform in size, and hence subtraction is not possible. The two matrices have different row and column counts.

Step 3: Conclusion

The operation \( C - 0.5D \) cannot be performed because matrix \( C \) and matrix \( D \) have different dimensions - 2x2 and 3x3, respectively. For matrix subtraction, the matrices must be of the same dimensions.

Key Concepts

Matrix Subtraction

Matrix subtraction is a fundamental operation in linear algebra, much like subtraction in arithmetic. It involves taking two matrices and subtracting the corresponding elements from each other. Works like regular subtraction: element-wise. For example, if you have two matrices, say \( X \) and \( Y \), both being 2x2 matrices, the subtraction \( X - Y \) would result in a new matrix where each element at position \((i, j)\) is calculated as \(x_{ij} - y_{ij}\). Each of these subtractions occurs independently of one another, maintaining the structure of the original matrices.
However, matrix subtraction can only be executed if both matrices have the same dimensions. This requirement ensures that each element from the first matrix has a direct counterpart in the same location of the second matrix for a meaningful operation. This simplicity allows for straightforward computation, yet adherence to dimensional conformity is a strict necessity.

Matrix Dimensions

Matrix dimensions denote the size of a matrix, expressed in terms of the number of rows and columns it contains. For instance, a 2x2 matrix indicates a matrix with two rows and two columns. These dimensions provide the fundamental boundaries within which matrix operations can occur.

• First value refers to rows.
• Second value refers to columns.

Understanding matrix dimensions is essential for identifying how matrices interact within arithmetic operations like addition, subtraction, and multiplication. This is because certain operations have specific requirements regarding the matrix dimensions. Paying attention to these dimensions will inform you about what operations can be executed and which cannot, thereby forming the basis for practical application of matrix mathematics.

Matrix Conformability

Matrix conformability is a term used to indicate whether two or more matrices are "compatible" in terms of their dimensions for specific operations. When it comes to matrix subtraction, as in the given exercise, both matrices need to have identical dimensions. If one matrix is a 2x2 and another is a 3x3, they are not conformable for subtraction.
Matrix conformability is crucial because it ensures each element in the first matrix has a corresponding element in the second matrix; they must "line up" perfectly.

• Matching rows and columns is necessary.
• If dimensions differ, operations like subtraction cannot proceed.

This concept isn't just confined to subtraction. It also applies to matrix addition and plays a role in determining viability in more complex operations like multiplication, although with different rules. Thus, understanding conformability is a vital step before performing any matrix operation.