Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

(a) \( \begin{bmatrix} 38 & -11 & 35 \end{bmatrix} \); (b) cannot be performed.

Step 1: Check Dimensions for Operation (a)

To perform the matrix operation \( D B + D C \), we need to verify if the matrices \( D \), \( B \), and \( C \) can be multiplied with each other. The matrix \( D \) has dimensions \( 1 \times 2 \), \( B \) has dimensions \( 2 \times 3 \), and \( C \) has dimensions \( 2 \times 3 \). The multiplication \( D B \) is possible because the number of columns in \( D \) (2) matches the number of rows in \( B \) (2), resulting in a \( 1 \times 3 \) matrix. Similarly, \( D C \) results in a \( 1 \times 3 \) matrix. Thus, we can perform both operations and add the resulting matrices.

Step 2: Calculate \( D B \)

Perform the multiplication of \( D = \begin{bmatrix} 7 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} \):\[ D B = \begin{bmatrix} 7 & 3 \end{bmatrix} \times \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} = \begin{bmatrix} 7 \times 3 + 3 \times 1 & 7 \times \frac{1}{2} + 3 \times (-1) & 7 \times 5 + 3 \times 3 \end{bmatrix} \]\[ = \begin{bmatrix} 21 + 3 & \frac{7}{2} - 3 & 35 + 9 \end{bmatrix} = \begin{bmatrix} 24 & \frac{1}{2} & 44 \end{bmatrix} \]

Step 3: Calculate \( D C \)

Perform the multiplication of \( D = \begin{bmatrix} 7 & 3 \end{bmatrix} \) and \( C = \begin{bmatrix} 2 & -\frac{5}{2} & 0 \ 0 & 2 & -3 \end{bmatrix} \):\[ D C = \begin{bmatrix} 7 & 3 \end{bmatrix} \times \begin{bmatrix} 2 & -\frac{5}{2} & 0 \ 0 & 2 & -3 \end{bmatrix} = \begin{bmatrix} 7 \times 2 + 3 \times 0 & 7 \times (-\frac{5}{2}) + 3 \times 2 & 7 \times 0 + 3 \times (-3) \end{bmatrix} \]\[ = \begin{bmatrix} 14 & -\frac{35}{2} + 6 & -9 \end{bmatrix} = \begin{bmatrix} 14 & -\frac{23}{2} & -9 \end{bmatrix} \]

Step 4: Perform \( D B + D C \)

Add the matrices obtained from the two calculations:\[ D B + D C = \begin{bmatrix} 24 & \frac{1}{2} & 44 \end{bmatrix} + \begin{bmatrix} 14 & -\frac{23}{2} & -9 \end{bmatrix} \]\[ = \begin{bmatrix} 24 + 14 & \frac{1}{2} - \frac{23}{2} & 44 - 9 \end{bmatrix} = \begin{bmatrix} 38 & -\frac{22}{2} & 35 \end{bmatrix} = \begin{bmatrix} 38 & -11 & 35 \end{bmatrix} \]

Step 5: Check Dimensions for Operation (b)

To perform the operation \( B F + F E \), check dimensions. \( B \) is \( 2 \times 3 \), \( F \) is \( 3 \times 3 \), and \( E \) is a \( 3 \times 1 \) column vector. The operation \( B F \) is valid and results in a \( 2 \times 3 \) matrix since the number of columns in \( B \) matches the number of rows in \( F \). However, the addition \( B F + F E \) is not possible because \( B F \) results in \( 2 \times 3 \) while \( F E \) results in a \( 3 \times 1 \) matrix. These two matrices cannot be added due to differing dimensions.

Key Concepts

Matrix Multiplication

Matrix multiplication is one of the fundamental operations in linear algebra where two matrices are combined to produce another matrix. The core concept here is that the number of columns in the first matrix must match the number of rows in the second matrix. This is crucial because for each element of the resultant matrix, a dot product is computed of a row from the first matrix and a column from the second. * For example, consider matrices \(D \) and \(B\) from our exercise. Matrix \(D\) is \(1 \times 2\) and \(B\) is \(2 \times 3\). Since the columns in \(D\) match the rows in \(B\), we can multiply these matrices. The result will be a \(1 \times 3\) matrix. * As demonstrated, the multiplication of \(D = \begin{bmatrix} 7 & 3 \end{bmatrix}\) by \(B\) results in \( \begin{bmatrix} 24 & \frac{1}{2} & 44 \end{bmatrix} \). For successful multiplication, keep in mind:

• The resulting matrix's dimensions will be determined by the outer dimensions (i.e., the number of rows in the first matrix and the number of columns in the second).
• Each element in the resulting matrix is a sum of products of corresponding entries from the row of the first matrix and the column of the second matrix.

Understanding how multiplication affects matrix dimensions and practicing calculating dot products can equip you to tackle any problem involving this operation.

Matrix Addition

Matrix addition is another simple yet powerful tool in matrix operations. To add two matrices, their dimensions must be identical. Specifically, they must have the same number of rows and columns. This requirement arises because addition is performed element-wise; each element of a matrix is added to its corresponding element in the other matrix. * In operation \(D B + D C\), both result in the same dimensions \((1 \times 3)\), which means they are valid for addition. * Thus, element-wise addition was performed as follows: \( \begin{bmatrix} 24 & \frac{1}{2} & 44 \end{bmatrix} + \begin{bmatrix} 14 & -\frac{23}{2} & -9 \end{bmatrix} \) to give \( \begin{bmatrix} 38 & -11 & 35 \end{bmatrix} \).Remember the following when working with matrix addition:

• Matrices must have the same dimensions.
• Corresponding elements are added together to create the resulting matrix.

Knowing these basics about matrix addition will help you effortlessly combine matrices when they meet the dimension criteria.

Dimensions of Matrices

Understanding the dimensions of matrices is essential when performing any operation involving matrices. Matrix dimensions are described in terms of rows and columns, formatted as \(m \times n\), where \(m\) indicates the number of rows and \(n\) indicates the number of columns. * For example, a matrix \(B\) with dimensions \(2 \times 3\) has 2 rows and 3 columns. * Identifying the dimensions is a preparatory step necessary before performing matrix operations like multiplication or addition.Why do dimensions matter? * They dictate which operations are possible. For multiplication, the number of columns in the first matrix must match the number of rows in the second matrix.* For addition, matrices must have the exact same dimensions.To ensure you correctly leverage matrix operations, always check the dimensions first. Not only does this allow for a smooth operation, but it also prevents incorrect operations that are mathematically invalid. Thus, understanding and identifying dimensions is a baseline skill in handling matrices effectively.