Problem 12 - Precalculus

Question

Use the matrices \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] \quad B=\left[\begin{array}{rr}0 & -3 \\ -5 & 2\end{array}\right] \quad C=\left[\begin{array}{rrr}10 & -\frac{11}{2} & 0 \\ \frac{3}{5} & 5 & 9\end{array}\right]\) \(D=\left[\begin{array}{rr}7 & -13 \\ -\frac{4}{3} & 0 \\\ 6 & 8\end{array}\right] \quad E=\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & 4 & -9 \\ 0 & 0 & -5\end{array}\right]\) to compute the following or state that the indicated operation is undefined. $$ E D $$

Step-by-Step Solution

Answer

The matrix multiplication \(ED\) results in the matrix: \[\begin{bmatrix} \frac{67}{3} & 11 \\ \frac{-178}{3} & -72 \\ -30 & -40 \end{bmatrix}\].

Step 1: Determine Compatibility for Multiplication

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.Matrix \(E\) has dimensions \(3 \times 3\) and matrix \(D\) has dimensions \(3 \times 2\). The number of columns in \(E\) is equal to the number of rows in \(D\), so the multiplication \(ED\) is valid.

Step 2: Set Up the Multiplication

The resulting matrix from multiplying a \(3 \times 3\) matrix by a \(3 \times 2\) matrix will have dimensions \(3 \times 2\).Therefore, the resulting matrix \(ED\) will have 3 rows and 2 columns.

Step 3: Perform the Multiplication for Each Element

Each element of the resulting matrix is calculated by taking the dot product of the corresponding row from \(E\) and the column from \(D\).For example, the element at \([1,1]\) in \(ED\) is calculated as:\[1 \times 7 + 2 \times \left(-\frac{4}{3}\right) + 3 \times 6 = 7 - \frac{8}{3} + 18 = 25 - \frac{8}{3} = \frac{75}{3} - \frac{8}{3} = \frac{67}{3}\]

Step 4: Compute All Elements of the Matrix

Continue the calculations for each element:- For element \([1,2]\):\(1 \times (-13) + 2 \times 0 + 3 \times 8 = -13 + 0 + 24 = 11\)- For element \([2,1]\):\(0 \times 7 + 4 \times \left(-\frac{4}{3}\right) + (-9) \times 6 = 0 - \frac{16}{3} - 54 = -54 - \frac{16}{3} = \frac{-162}{3} - \frac{16}{3} = \frac{-178}{3}\)- For element \([2,2]\):\(0 \times (-13) + 4 \times 0 + (-9) \times 8 = 0 + 0 - 72 = -72\)- For element \([3,1]\):\(0 \times 7 + 0 \times \left(-\frac{4}{3}\right) + (-5) \times 6 = 0 + 0 - 30 = -30\)- For element \([3,2]\):\(0 \times (-13) + 0 \times 0 + (-5) \times 8 = 0 + 0 - 40 = -40\)

Step 5: Write the Resultant Matrix

Compile all calculated elements into the \(3 \times 2\) matrix:\[ED = \begin{bmatrix}\frac{67}{3} & 11 \\frac{-178}{3} & -72 \-30 & -40\end{bmatrix}\]This is the final result for the matrix multiplication \(ED\).

Key Concepts

Dot Product

The dot product is a fundamental calculation in matrix multiplication. It involves multiplying corresponding elements from a row of the first matrix by a column of the second matrix, and then summing these products.

For example, consider the element in the first row and first column of the product matrix. This value comes from the dot product of the first row of matrix \(E\) and the first column of matrix \(D\): \[1 \times 7 + 2 \times \left(-\frac{4}{3}\right) + 3 \times 6 = \frac{67}{3}.\] Each element in the resultant matrix is calculated using this method, highlighting the dot product's role in matrix multiplication. It's important because it systematically leads to filling up the entirety of the new matrix.

Matrix Dimensions

Understanding matrix dimensions is essential for performing various matrix operations, including matrix multiplication. A matrix’s dimensions are given in terms of its rows and columns. For instance, if a matrix has 3 rows and 2 columns, it is termed a \(3 \times 2\) matrix.

In our example, matrix \(E\) is described as having dimensions \(3 \times 3\), meaning it has 3 rows and 3 columns. Matrix \(D\), on the other hand, is a \(3 \times 2\) matrix.

Knowing these dimensions helps in understanding whether matrix multiplication is possible (more on that under multiplication compatibility). Regardless of the matrix operation you are doing, keeping dimensions in mind ensures you clearly identify and manage the data.

Matrix Operations

Matrix operations encompass several calculations you can perform with matrices, such as addition, subtraction, and multiplication. Each of these operations has its rules. Multiplication, for instance, requires care with dimensions and calculating dot products for each element in the resulting matrix.

The product matrix dimensions, when multiplying two matrices, are determined by the number of rows in the first matrix and the number of columns in the second. For the matrices \(E\ (3 \times 3)\) and \(D\ (3 \times 2)\), the result is then a \(3 \times 2\) matrix.

• Addition and subtraction require matrices of the same dimension.
• Multiplication needs compatible dimensions (columns of the first equal rows of the second).

Appropriate calculation methods for each operation ensure accuracy and proper formation of resultant matrices.

Multiplication Compatibility

A key concept in matrix multiplication is ensuring multiplication compatibility. This means the number of columns in the first matrix must match the number of rows in the second matrix. Only then is multiplication possible.

For instance, with matrices \(E\) and \(D\):

• The first matrix, \(E\), has \(3\) columns.
• The second matrix, \(D\), has \(3\) rows.

Thus, these matrices meet the compatibility requirement, which is why their multiplication \(ED\) is defined and can produce a valid \(3 \times 2\) matrix.

Checking for compatibility is an essential early step to ensure matrix operations can be carried out effectively, preventing errors and incorrect solutions.