Problem 28 - College Algebra

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. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {6} & {7}\end{array}\right], B=\left[\begin{array}{cc}{-9} & {6} \\ {-4} & {2}\end{array}\right], C=\left[\begin{array}{cc}{0} & {9} \\ {7} & {1}\end{array}\right], D=\left[\begin{array}{rrr}{-8} & {7} & {-5} \\ {4} & {3} & {2} \\ {0} & {9} & {2}\end{array}\right], E=\left[\begin{array}{rrr}{4} & {5} & {3} \\ {7} & {-6} & {-5} \\ {1} & {0} & {9}\end{array}\right] $$ $$ C-0.5 D $$

Step-by-Step Solution

Answer

The operation is not possible because the matrices have different dimensions.

Step 1: Identify the Operation to Be Performed

The given operation is to subtract matrix \(0.5D\) from matrix \(C\). This means we need to scale the matrix \(D\) by 0.5 and then perform matrix subtraction with \(C\).

Step 2: Check Matrix Dimensions

Matrix \(C\) is a 2x2 matrix, and matrix \(D\) is a 3x3 matrix. For matrix subtraction to be possible, both matrices must be of the same dimensions. Thus, subtracting \(0.5D\) from \(C\) is not possible because their dimensions do not match.

Step 3: Conclude the Operation

Since the matrices \(C\) and \(0.5D\) are not of the same size, the operation \(C - 0.5D\) cannot be performed. It is essential that both matrices involved in a subtraction have identical rows and columns, which is not the case here.

Key Concepts

Matrix Subtraction

Matrix subtraction involves taking two matrices and subtracting the corresponding elements. For two matrices to be subtracted, they must be of the same dimensions. This means the number of rows in both matrices has to be equal, and the number of columns as well.

Here's why it's important:

• Each element in one matrix is directly subtracted from the corresponding element in the other matrix. For instance, if you have element \(a_{11}\) in matrix A and \(b_{11}\) in matrix B, their subtraction would result in \(a_{11} - b_{11}\).
• Different dimensions make direct element-to-element subtraction impossible. Hence, the dimensions of the two matrices must align perfectly.

Trying to subtract matrices with different dimensions will inevitably lead to an operation that can't be performed in matrix arithmetic. Therefore, checking the dimensions beforehand is a crucial step.

Scalar Multiplication

Scalar multiplication is where each element of a matrix is multiplied by a scalar (a single number). It's a straightforward but essential operation used often in matrix algebra.

Here's how it works:

• You take a scalar value, like 0.5 in our problem, and multiply it with each element of the matrix.
• This influences all parts of the matrix equally. Thus, each original element becomes a product of itself and the scalar.

For example, if you have matrix D and multiply it by 0.5, you'd find each element of D halved. So, \(D\)'s element \(d_{11}\) becomes \(0.5 \, d_{11}\), and so on. Scalar multiplication is crucial when you need to scale matrices up or down without altering their dimensions.

Matrix Dimensions

Understanding matrix dimensions is vital before performing any operations, as they dictate possible arithmetic actions between matrices. Each matrix is denoted by the number of rows followed by the number of columns, usually expressed as \(m \, \text{x} \, n\).

Here's why they matter:

• They determine the compatibility for operations like addition and subtraction. The dimensions must be identical for these operations.
• They indicate how a matrix can interact with another matrix or a scalar. For instance, a 2x2 matrix interacts differently than a 2x3 matrix due to dimension constraints.

In the exercise, matrix C has a dimension of 2x2 and matrix D is 3x3. Their differing dimensions make subtraction impossible. Thus, always knowing the dimensions will guide the operations you can perform.