Problem 18 - Precalculus: Mathematical for Calculus

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as $$\begin{array}{l} A=\left[\begin{array}{rr} 2 & -5 \\ 0 & 7 \end{array}\right] \quad B=\left[\begin{array}{rrrr} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array}\right] \quad C=\left[\begin{array}{rrr} 2 & -\frac{5}{2} & 0 \\ 0 & 2 & -3 \end{array}\right] \\ D=\left[\begin{array}{rrr} 7 & 3 \end{array}\right] & E=\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right] \\ F=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \quad G=\left[\begin{array}{rrr} 5 & -3 & 10 \\ 6 & 1 & 0 \\ -5 & 2 & 2 \end{array}\right] \end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$B+F$$

Step-by-Step Solution

Answer

The operation \(B + F\) cannot be performed because the matrices have different dimensions.

Step 1: Determine Matrix Dimensions

First, identify the dimensions of the matrices. Matrix \(B\) has dimensions \(2 \times 3\) as it has 2 rows and 3 columns. Matrix \(F\) is a \(3 \times 3\) matrix, with 3 rows and 3 columns.

Step 2: Check for Matrix Addition Compatibility

In order to add two matrices together, they must have identical dimensions. Compare the dimensions of matrices \(B\) and \(F\). Matrix \(B\) with dimensions \(2 \times 3\) and matrix \(F\) with dimensions \(3 \times 3\) do not match.

Step 3: Conclusion on Matrix Addition

Since matrices \(B\) and \(F\) have different dimensions (\(2 \times 3\) versus \(3 \times 3\)), the addition \(B + F\) cannot be performed. Therefore, matrix addition is not possible for these matrices.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is essential when working with matrices. The dimensions of a matrix are described by two numbers: the number of rows and the number of columns. This is usually written in the form "rows × columns". For instance, if a matrix has 2 rows and 3 columns, its dimensions are written as "2 × 3." Each element of the matrix is addressed using these row and column indices, like cell addresses in a spreadsheet.

To visualize this, imagine a matrix as a grid of numbers. The rows run horizontally and the columns vertically. For the given matrices in our exercise:

• Matrix B is a 2 × 3 matrix, meaning it has 2 rows and 3 columns.
• Matrix F is a 3 × 3 matrix, with 3 rows and 3 columns.

When primarily dealing with matrices, keeping track of their dimensions is vital for performing operations like addition or multiplication.

Matrix Compatibility

Matrix compatibility is crucial when performing operations such as addition or multiplication. For matrix addition, the matrices need to have the exact same dimensions. This means they need an identical number of rows and columns.

In our exercise, the dimensions of Matrix B are 2 × 3 and those of Matrix F are 3 × 3. Since these dimensions are not identical, the two matrices are incompatible for addition. In simple terms, you can't add them because their shapes don't match up.

If you picture trying to place a 2 × 3 piece on top of a 3 × 3 piece, it will not completely cover it; therefore, they don't align correctly for addition. So for successful matrix addition, always ensure your matrices are compatible by having matching dimensions.

Matrix Operations

Matrix operations encompass various methods used to manipulate matrices. Common operations include addition, subtraction, and multiplication, each with specific rules.

For matrix addition, corresponding elements from each matrix must be added together. This is only possible if both matrices have the same dimensions — they must be compatible, as described in the previous section. Thus, in our exercise, attempting to add the matrices B and F, with dimensions 2 × 3 and 3 × 3 respectively, is not feasible.

Understanding these operations requires knowing the underlying rules:

• Addition: Compatible matrices (same dimensions) have corresponding elements added together.
• Subtraction: Similar to addition, but elements from one matrix are subtracted by corresponding elements of the other.
• Multiplication: More complex, involving the dot product of rows and columns, where the number of columns of the first matrix must match the number of rows of the second.

Each operation has its importance in solving matrix equations and requires careful validation of dimensions before proceeding.