In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason. 4 0 5 1 1 - 1 A = - 3 5 B = C = 0 1 - 2 - 2 - 1 1 A - C

Verified step by step guidance

1. Step 1: Identify the dimensions of matrices A and C. Matrix A is a 3x2 matrix (3 rows, 2 columns), and matrix C is a 2x2 matrix (2 rows, 2 columns).
2. Step 2: Recall that matrix subtraction A - C is only defined if both matrices have the same dimensions. Since A is 3x2 and C is 2x2, their dimensions do not match.
3. Step 3: Conclude that the operation A - C is not defined because the matrices have different dimensions and cannot be subtracted element-wise.
4. Step 4: No further calculations are needed since the operation is undefined.
5. Step 5: State the reason clearly: Matrix subtraction requires matrices to be of the same size, which is not the case here.

Key Concepts

Matrix Addition and Subtraction

Matrix addition and subtraction involve combining matrices by adding or subtracting their corresponding elements. These operations are only defined when the matrices have the same dimensions. For example, to compute A - C, both A and C must be of the same size.

Matrix Dimensions and Compatibility

The dimensions of a matrix are given by its number of rows and columns. Operations like addition, subtraction, and multiplication require specific dimension compatibility. For addition and subtraction, matrices must have identical dimensions; otherwise, the operation is undefined.

Matrix Representation and Notation

Matrices are rectangular arrays of numbers arranged in rows and columns, typically denoted by uppercase letters. Understanding how to read and interpret matrix elements is essential for performing operations correctly, such as identifying elements by their row and column positions.