Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result of \(DB + DC\) is \(\left[38, -11, 35\right]\).

Step 1: Verify Matrix Multiplication Compatibility for DB

The matrix \(D\) is \(1 \times 2\), and the matrix \(B\) is \(2 \times 3\). For matrix multiplication to be possible, the number of columns in \(D\) (which is 2) must match the number of rows in \(B\) (which is 2). Since they match, the multiplication \(DB\) is possible and will result in a \(1 \times 3\) matrix.

Step 2: Calculate DB

Multiply matrix \(D\) by matrix \(B\). \[ DB = \left[ \begin{array}{ll} 7 & 3 \end{array} \right] \left[ \begin{array}{rrr} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{array} \right] = \left[ \begin{array}{lll} 7 \cdot 3 + 3 \cdot 1 & 7 \cdot \frac{1}{2} + 3 \cdot (-1) & 7 \cdot 5 + 3 \cdot 3 \end{array} \right] \]\[ = \left[ \begin{array}{lll} 21 + 3 & 3.5 - 3 & 35 + 9 \end{array} \right] = \left[ \begin{array}{lll} 24 & 0.5 & 44 \end{array} \right] \]

Step 3: Verify Matrix Multiplication Compatibility for DC

The matrix \(D\) is \(1 \times 2\), and the matrix \(C\) is \(2 \times 3\). For matrix multiplication to be possible, the number of columns in \(D\) (which is 2) must match the number of rows in \(C\) (which is 2). Since they match, the multiplication \(DC\) is possible and will result in a \(1 \times 3\) matrix.

Step 4: Calculate DC

Multiply matrix \(D\) by matrix \(C\). \[ DC = \left[ \begin{array}{ll} 7 & 3 \end{array} \right] \left[ \begin{array}{rrr} 2 & -\frac{5}{2} & 0 \ 0 & 2 & -3 \end{array} \right] = \left[ \begin{array}{lll} 7 \cdot 2 + 3 \cdot 0 & 7 \cdot (-\frac{5}{2}) + 3 \cdot 2 & 7 \cdot 0 + 3 \cdot (-3) \end{array} \right] \]\[ = \left[ \begin{array}{lll} 14 & -\frac{35}{2} + 6 & -9 \end{array} \right] = \left[ \begin{array}{lll} 14 & -17.5 + 6 & -9 \end{array} \right] = \left[ \begin{array}{lll} 14 & -11.5 & -9 \end{array} \right] \]

Step 5: Add Matrices DB and DC

To add the results of \(DB\) and \(DC\), make sure both are the same size. Both are \(1 \times 3\) matrices: \[ DB + DC = \left[ \begin{array}{lll} 24 & 0.5 & 44 \end{array} \right] + \left[ \begin{array}{lll} 14 & -11.5 & -9 \end{array} \right] \]Add corresponding elements: \[ = \left[ \begin{array}{lll} 24 + 14 & 0.5 - 11.5 & 44 - 9 \end{array} \right] = \left[ \begin{array}{lll} 38 & -11 & 35 \end{array} \right] \]

Step 6: State Final Answer

Thus, the result of the operation \(DB + DC\) is the matrix \(\left[ \begin{array}{lll} 38 & -11 & 35 \end{array} \right]\).

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra. It involves multiplying two matrices to produce a new matrix. For matrix multiplication to occur, an important rule must be adhered to. The number of columns in the first matrix must match the number of rows in the second matrix. This compatibility ensures that each element from the rows of the first matrix can "pair" with an element in the columns of the second matrix.

For example, when multiplying matrix \( D \) of size \( 1 \times 2 \) with matrix \( B \) of size \( 2 \times 3 \), the number of columns in \( D \) (which is 2) matches the number of rows in \( B \) (also 2). Hence, they are compatible, and their multiplication results in a \( 1 \times 3 \) matrix.

The resulting matrix will have elements calculated by multiplying the corresponding elements and summing them up for each position in the resulting matrix. This process may initially seem complex, but with practice, it becomes intuitive and straightforward.

Matrix Addition

Matrix addition is a simpler operation compared to multiplication. Two matrices can be added together only if they have the same dimensions. It means they must have the same number of rows and columns. The addition process involves taking corresponding elements from each matrix and simply summing them up.

For instance, once we have multiplied \( D \) by both \( B \) and \( C \), we end up with two \( 1 \times 3 \) matrices. Since these matrices are identical in dimension, we can add them together. Each element of the resulting matrix is obtained by adding the corresponding elements of the matrices \( DB \) and \( DC \). If you ensure matrices have matching sizes, matrix addition remains an efficient and clear operation.

Matrix Compatibility

Matrix compatibility plays a fundamental role in determining whether certain operations, like multiplication and addition, are feasible. Compatibility rules ensure that operations performed on matrices are valid, adhering strictly to mathematical conventions.

• For multiplication, the inner dimensions -- the columns of the first matrix and the rows of the second matrix -- must match. This rule ensures every element in a row pairs uniquely with an element from a column.


• For addition, matrices should be exactly the same size, both in terms of number of rows and columns. Without this equality, addition isn't defined.

Understanding compatibility is key to mastering matrix operations, as it prevents undefined operations and errors during calculations. By checking compatibility rules beforehand, we can proceed confidently with matrix operations.

Algebraic Operations

In algebra, operations on matrices enhance the ability to solve complex equations and evaluate expressions involving multiple variables and parameters. Algebraic operations on matrices, such as addition, subtraction, and multiplication, allow for the manipulation and transformation of data.

When carrying out a sequence of algebraic operations, such as the expression \( DB + DC \), it is crucial to follow the order of operations. First, matrix multiplications are executed, followed by addition or subtraction only if the resulting matrices from the previous operations have compatible dimensions.

Embarking on algebraic operations within matrices requires a systematic approach, adhering to compatibility rules, and ensuring that each step logically follows from the last. By mastering these operations, students can solve matrix equations with confidence and clarity, paving the way for further studies in linear algebra.