Problem 26 - Precalculus

Question

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

Step-by-Step Solution

Answer

The matrix product \( BF \) results in \( \begin{bmatrix} -1 \\ -1 \end{bmatrix} \).

Step 1: Identify matrices' dimensions

Matrix \( B \) is a \( 2 \times 3 \) matrix and matrix \( F \) is a \( 3 \times 1 \) column matrix.

Step 2: Verify if matrix multiplication is possible

Matrix multiplication \( BF \) is possible if the number of columns in \( B \) is equal to the number of rows in \( F \). Since \( B \) has 3 columns and \( F \) has 3 rows, they can be multiplied.

Step 3: Set up the multiplication

Position matrix \( B \) as \( \left[ \begin{array}{ccc} 0 & 2 & 1 \ 3 & -2 & 4 \end{array} \right] \) and matrix \( F \) as \( \left[ \begin{array}{c} 1 \ 0 \ -1 \end{array} \right] \).

Step 4: Compute the resulting matrix

The resulting matrix will have dimensions equal to the number of rows of \( B \) and the number of columns of \( F \), giving us a \( 2 \times 1 \) matrix. Calculate each element:- First row element: \( 0 \times 1 + 2 \times 0 + 1 \times (-1) = -1 \).- Second row element: \( 3 \times 1 + (-2) \times 0 + 4 \times (-1) = 3 - 4 = -1 \).

Step 5: Write down the resulting matrix

Combine the results from the computations:The resulting matrix is: \[BF = \begin{bmatrix} -1 \ -1 \end{bmatrix}.\]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial before performing any matrix operation. A matrix is defined by its dimensions, often denoted as 'rows x columns'. For instance, if a matrix has two rows and three columns, it is described as a \(2 \times 3\) matrix. Knowing the dimensions is important because many matrix operations, such as multiplication, depend on compatible dimensions.

Before attempting to multiply matrices, always identify their dimensions. This helps determine if the operation is feasible. In our example, matrix \(B\) is a \(2 \times 3\) matrix, which means it has 2 rows and 3 columns, while matrix \(F\) is a \(3 \times 1\) matrix, having 3 rows and 1 column. These dimensions will play a critical role in matrix multiplication.

Matrix Operations

Matrix operations include tasks such as addition, subtraction, and multiplication. Unlike basic arithmetic, these operations have their own set of rules and limitations. For instance, matrix multiplication requires careful attention to dimension compatibility.

When preparing to execute these operations, particularly multiplication, check that the matrices are dimensionally compatible. This means ensuring the number of columns in the first matrix matches the number of rows in the second. Other operations, like addition and subtraction, require matrices to have identical dimensions. Therefore, always verify each matrix pair's dimensional suitability before proceeding with an operation.

In our worked example, the task involved the multiplication of matrices \(B\) and \(F\), which was possible due to the compatible dimensions.

Matrix Multiplication Rules

Matrix multiplication follows specific rules that differ from regular number multiplication. A crucial requirement is that the number of columns in the first matrix must equal the number of rows in the second matrix. This condition enables the dot product computation of corresponding elements.

Here's the process for performing matrix multiplication:

• Take an element from a row of the first matrix and multiply it with the corresponding element in a column of the second matrix.
• Add these products to get the final element for the resulting matrix.

In our example, matrix \(B\) (\(2 \times 3\)) and matrix \(F\) (\(3 \times 1\)) were multiplied to yield a \(2 \times 1\) matrix. This result was achieved by calculating the dot products of \(B\)'s rows and \(F\)'s column. This process shows that the dimensions of the resulting matrix are derived from the outer dimensions of the multiplied matrices. Ensuring these rules are followed ensures a smooth calculation and an accurate understanding of matrix multiplication.