Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

4B - 3C

Verified step by step guidance

1. Step 1: Identify the matrices and the operation to be performed. We are asked to compute the expression \$4B - 3C\(, where matrices \)B\( and \)C\( are given as: \)B = \begin{bmatrix} 5 & 1 \\ -2 & -2 \end{bmatrix}\( and \)C = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$.
2. Step 2: Multiply matrix \(B\) by the scalar 4. This means multiplying each element of \(B\) by 4: \$4B = 4 \times \begin{bmatrix} 5 & 1 \\ -2 & -2 \end{bmatrix} = \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. This means multiplying each element of \(C\) by 3: \$3C = 3 \times \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} = \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\(. This involves subtracting corresponding elements: \)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 are \$2 \times 2$ matrices, the subtraction is defined and can be performed element-wise.

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. It is only defined when the number of columns in the first matrix equals the number of rows in the second. This operation is not commutative, meaning AB does not necessarily equal BA.

Scalar Multiplication of Matrices

Scalar multiplication involves multiplying every element of a matrix by a constant (scalar). This operation changes the magnitude of the matrix elements but not the matrix dimensions. It is distributive over matrix addition and associative with scalar multiplication.

Matrix Dimensions and Operation Validity

The dimensions of matrices determine which operations are defined. For 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. Understanding dimensions helps identify if an operation is possible.