Problem 35 - Precalculus

Question

perform the indicated operations for each expression, if possible. $$A=\left[\begin{array}{rrr}-1 & 3 & 0 \\\2 & 4 & 1\end{array}\right] \quad B=\left[\begin{array}{rr}0 & 2 & 1 \\\3 & -2 & 4\end{array}\right] \quad C=\left[\begin{array}{rr}0 & 1 \\\2 & -1 \\\3 & 1 \end{array}\right] \quad D=\left[\begin{array}{rr}2 & -3 \\\0 & 1 \\\4 & -2 \end{array}\right]$$ $$E=\left[\begin{array}{rrr}-1 & 0 & 1 \\\2 & 1 & 4 \\\\-3 & 1 & 5 \end{array}\right] \quad F=\left[\begin{array}{r}1 \\\0 \\\\-1\end{array}\right] \quad G=\left[\begin{array}{ll}1 & 2 \\\3 & 4\end{array}\right]$$ $$B(A+E)$$

Step-by-Step Solution

Answer

The operation \( B(A+E) \) is undefined because matrices A and E cannot be added due to different dimensions.

Step 1: Perform the fraction operation

Apply the rules for adding, subtracting, multiplying, or dividing fractions.

Step 2: Simplify

The simplified result is The operation \( B(A+E) \) is undefined because matrices A and E cannot be added due to different di.

Key Concepts

Matrix Addition

Matrix addition is a relatively straightforward operation, but it comes with specific rules that must be followed. To add two matrices, they must have the same dimensions. This means each matrix must have the same number of rows and columns. For instance, if you have Matrix A with dimensions 2x3, you can only add it to another 2x3 matrix. Each element in one matrix is added to the corresponding element in the other. This element-by-element addition creates a new matrix of the same dimensions.

• Example: If Matrix P = \ \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) and Matrix Q = \ \( \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} \), then their sum R = \ \( \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix} \).

It's important to check dimensions before adding matrices to ensure the operation is possible.

Matrix Multiplication

Matrix multiplication is a bit more complex than matrix addition. In multiplication, the dimensions of the matrices are crucially important. You can only multiply two matrices if the number of columns in the first matrix matches the number of rows in the second. Let's say you have a matrix X of dimensions 2x3 and a matrix Y of dimensions 3x2. The product of these matrices will be a new matrix Z with dimensions 2x2.

• The entry at the first row and first column in Matrix Z is calculated by multiplying the elements of the first row in Matrix X by the corresponding elements of the first column in Matrix Y and summing up the products.

This pattern continues for each element in the resulting matrix. This is why understanding the rules of dimensions is so crucial for performing matrix multiplication.

Dimensions of Matrices

The dimensions of a matrix are defined by the number of rows and columns it contains, typically expressed as 'rows x columns'. For example, a matrix A that has 2 rows and 3 columns is referred to as a 2x3 matrix. Understanding matrix dimensions is fundamental to performing various matrix operations like addition and multiplication. Each type of operation requires specific conditions regarding dimensions:

• For addition, matrices must have identical dimensions.
• For multiplication, the number of columns in the first matrix must equal the number of rows in the second.

Recognizing matrix dimensions is akin to noting the "shape" or "size" of the matrix, which determines compatibility for certain mathematical operations.

Undefined Operations

Operations on matrices can sometimes be undefined due to incompatible dimensions. In the problem at hand, the task was to evaluate the expression \( B(A+E) \). However, since matrices A and E have differing dimensions (A is 2x3 and E is 3x3), you cannot add them. As a result, the operation \( A+E \) is undefined, making the subsequent multiplication \( B(A+E) \) impossible.

• Each operation, whether addition or multiplication, has specific dimensional requirements that must be met to avoid undefined results.

Always check dimensions before attempting any matrix operation to determine if the procedure is feasible or if it's bound to be undefined.