Problem 18 - Discrete Mathematics with Applications

Question

Let \(A=\left[\begin{array}{lll}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{rrr}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{rrr}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right] .\) Find each. $$A(B+C)$$

Step-by-Step Solution

Answer

The final answer is \(A(B+C) = \begin{bmatrix} 3 &-1 & 0 \\ 3 & 11 & 13 \end{bmatrix}\).

Step 1: Find the sum of matrices B and C

To find the sum of two matrices, we add each corresponding element from the matrices B and C. So, \[B+C = \begin{bmatrix} 0 & -2 & 5 \\ 0 & 0 & 1 \end{bmatrix} + \begin{bmatrix} -3 & 0 & 0 \\ 0 & 1 & 2 \end{bmatrix}\]

Step 2: Add corresponding elements

Add elements from each matrices entry-wise. \[(B+C)_{i,j} = B_{i,j}+C_{i,j}\] So, \[ B+C = \begin{bmatrix} 0 + (-3) & -2 + 0 & 5 + 0 \\ 0 + 0 & 0 + 1 & 1 + 2 \end{bmatrix} \]

Step 3: Simplify the sum

Simplify the resulting matrix: \[ B+C = \begin{bmatrix} -3 & -2 & 5 \\ 0 & 1 & 3 \end{bmatrix} \]

Step 4: Multiply matrix A with the sum (B+C)

Now we multiply matrix A by the sum obtained in the previous step: \[A(B+C) = \begin{bmatrix} 1 & 0 & -1 \\ 0 & 2 & 3 \end{bmatrix} \begin{bmatrix} -3 & -2 & 5 \\ 0 & 1 & 3 \end{bmatrix}\]

Step 5: Perform matrix multiplication

Use the definition of matrix multiplication: \[(AB)_{i,j} = \sum_{k=1}^{n} A_{i,k} \cdot B_{k,j}\] So, the resulting matrix, with each element of the new matrix being the product of the corresponding row elements of A and column elements of B+C, looks like: \[ A(B+C) = \begin{bmatrix} (1 \cdot -3) + (0 \cdot 0) + (-1 \cdot 5) & (1 \cdot -2) + (0 \cdot 1) + (-1 \cdot 3) & (1 \cdot 5) + (0 \cdot 3) + (-1 \cdot 5) \\ (0 \cdot -3) + (2 \cdot 0) + (3 \cdot 1) & (0 \cdot -2) + (2 \cdot 1) + (3 \cdot 3) & (0 \cdot 5) + (2 \cdot 3) + (3 \cdot 3) \end{bmatrix} \]

Step 6: Simplify the resulting matrix

Simplify each entry of the matrix: \[ A(B+C) = \begin{bmatrix} 3 &-1 & 0 \\ 3 & 11 & 13 \end{bmatrix} \] The final answer is \(A(B+C) = \begin{bmatrix} 3 &-1 & 0 \\ 3 & 11 & 13 \end{bmatrix}\).

Key Concepts

Matrix Addition

Matrix addition is a straightforward operation in linear algebra. It involves adding two matrices by summing up their corresponding elements. To perform matrix addition, two matrices must be of the same order, meaning they have the same number of rows and columns. Consider matrices \( B \) and \( C \) from the exercise. These matrices are both 2x3 matrices.

• To add these, just add each element in the same position in both matrices.
• For matrix \( B \), the element in the first row and first column is 0, while in matrix \( C \), it is -3.
• The sum is \( 0 + (-3) = -3 \) for this position. This process repeats for every element in the matrices.

Matrix addition is as simple as computing the sum in this manner for all positions. As a result, the new matrix is constructed where each element is the sum of the corresponding elements of the input matrices. In our example, the result of adding matrices \( B \) and \( C \) was \[B + C = \begin{bmatrix} -3 & -2 & 5 \ 0 & 1 & 3 \end{bmatrix}. \] Understanding this gateway to more complex operations is essential for mastering matrix manipulation.

Matrix Multiplication

Unlike addition, matrix multiplication is a bit more detailed. Here, the number of columns in the first matrix must equal the number of rows in the second. In the exercise, matrix \( A \) is multiplied by the matrix \( (B+C) \). Matrix \( A \) is a 2x3 matrix and \( (B+C) \) is also a 2x3 matrix. For them to be compatible for multiplication, both need to be arranged such that the condition is met. However, the exercise correctly uses appropriately arranged submatrices for multiplication.

• To multiply, take a row from the first matrix and a column from the second.
• Multiply each respective element pair in this row and column.
• Sum these products for each pair to get one element of the resulting matrix.

This operation is repeated across all elements to obtain the product matrix. Calculating the elements for a matrix like \( (A \cdot (B+C)) \) involves steps like:
\[(AB)_{1,1} = (1 \cdot -3) + (0 \cdot 0) + (-1 \cdot 5) = -3 + 0 - 5 = -8.\]
Despite the complex nature of this operation, practice enhances your understanding and makes it easier over time.

Elementary Matrices

Elementary matrices are special matrices that are used in computational problems to perform basic transformations on other matrices. They are constructed by applying a single elementary row operation to the identity matrix. These operations can be classified into three types:

• Swapping two rows,
• Multiplying a row by a non-zero scalar,
• Adding a multiple of one row to another row.

An understanding of elementary matrices is crucial because they serve as building blocks in matrix decompositions and play a role in solving systems of linear equations. By transforming matrices into row-echelon form, elementary matrices simplify the application of advanced matrix concepts like inverses or determinant calculation.
This exercise did not explicitly use elementary matrices, but principles of manipulation like these help build foundations in matrix operations. Mastery of these basic operations makes complex topics less daunting and enhances problem-solving efficiency in linear algebra.