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 4B - 3C

Verified step by step guidance

1. Step 1: Understand the matrices and the operation to perform. You are asked to compute the expression 4B - 3C, where B and C are given matrices.
2. Step 2: Multiply matrix B by the scalar 4. This means multiplying each element of matrix B by 4. If B = \( \begin{bmatrix} 5 & 1 \\ -2 & -2 \end{bmatrix} \), then 4B = \( \begin{bmatrix} 4 \times 5 & 4 \times 1 \\ 4 \times (-2) & 4 \times (-2) \end{bmatrix} \).
3. Step 3: Multiply matrix C by the scalar 3. Similarly, multiply each element of matrix C by 3. If C = \( \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \), then 3C = \( \begin{bmatrix} 3 \times 1 & 3 \times (-1) \\ 3 \times (-1) & 3 \times 1 \end{bmatrix} \).
4. Step 4: Subtract the matrix 3C from the matrix 4B. This means subtracting corresponding elements of 3C from 4B. The resulting matrix will be \( 4B - 3C = \begin{bmatrix} (4B)_{11} - (3C)_{11} & (4B)_{12} - (3C)_{12} \\ (4B)_{21} - (3C)_{21} & (4B)_{22} - (3C)_{22} \end{bmatrix} \).
5. Step 5: Verify the dimensions of matrices B and C are the same before performing the subtraction. Since both B and C are 2x2 matrices, the subtraction is defined and can be performed element-wise.

Key Concepts

Matrix Addition and Scalar Multiplication

Matrix addition involves adding corresponding elements of two matrices of the same dimensions. Scalar multiplication means multiplying every element of a matrix by a scalar (a real number). Both operations require matrices to have compatible dimensions to be defined.

Matrix Dimensions and Compatibility

The dimensions of a matrix are given by its number of rows and columns. For operations like addition or subtraction, matrices must have the same dimensions. For multiplication, the number of columns in the first matrix must equal the number of rows in the second.

Determining if an Operation is Defined

Before performing matrix operations, it is essential to check if the operation is defined based on matrix dimensions. If dimensions do not match the requirements for addition, subtraction, or multiplication, the operation is undefined and must be stated as such.