Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations. \(\left[\begin{array}{rrr}1 & 4 & -5 \\ 3 & -8 & 6\end{array}\right]+\left[\begin{array}{rrr}4 & 0 & -2 \\ 3 & 6 & 5\end{array}\right]-\left[\begin{array}{rrr}2 & 8 & 9 \\ -11 & 2 & -5\end{array}\right]\)

Step-by-Step Solution

Answer

The result of the indicated operations is: \[\left[\begin{array}{rrr}3 & -4 & -16 \\\ 17 & -4 & 16\end{array}\right]\]

Step 1: Add the first two matrices

To add two matrices, we simply add their corresponding elements. Add the first and the second matrices. \[\left[\begin{array}{rrr}1 & 4 & -5 \\\ 3 & -8 & 6\end{array}\right]+\left[\begin{array}{rrr}4 & 0 & -2 \\\ 3 & 6 & 5\end{array}\right] = \left[\begin{array}{rrr}1+4 & 4+0 & -5+(-2) \\\ 3+3 & -8+6 & 6+5\end{array}\right]\]

Step 2: Evaluate the sum

Evaluate the sum of the first two matrices. \[\left[\begin{array}{rrr}1+4 & 4+0 & -5+(-2) \\\ 3+3 & -8+6 & 6+5\end{array}\right] = \left[\begin{array}{rrr}5 & 4 & -7 \\\ 6 & -2 & 11\end{array}\right]\]

Step 3: Subtract the third matrix

Now subtract the third matrix from the sum of the first two matrices. \[\left[\begin{array}{rrr}5 & 4 & -7 \\\ 6 & -2 & 11\end{array}\right]-\left[\begin{array}{rrr}2 & 8 & 9 \\\ -11 & 2 & -5\end{array}\right] = \left[\begin{array}{rrr}5-2 & 4-8 & (-7)-9 \\\ 6-(-11) & (-2)-2 & 11-(-5)\end{array}\right]\]

Step 4: Evaluate the difference

Now evaluate the difference between the sum of the first two matrices and the third matrix. \[\left[\begin{array}{rrr}5-2 & 4-8 & (-7)-9 \\\ 6-(-11) & (-2)-2 & 11-(-5)\end{array}\right] = \left[\begin{array}{rrr}3 & -4 & -16\\\ 17 & -4 & 16\end{array}\right]\] The result of the indicated operations is: \[\left[\begin{array}{rrr}3 & -4 & -16 \\\ 17 & -4 & 16\end{array}\right]\]

Key Concepts

Matrix Addition

Matrix addition is one of the fundamental operations in linear algebra used to combine matrices by adding their respective elements. This operation is only possible when the matrices involved are "conformable," i.e., they have the same dimensions. For example, if you are asked to add two matrices, both must have the same number of rows and columns.

To add matrices,

• Take the first element of the first matrix and add it to the first element of the second matrix.
• Do this for each corresponding element in the matrices.

Let us visualize this. Consider two matrices, \[A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\]When we add these, we get:\[A + B = \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \ a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}\]Addition of matrices is both commutative and associative, meaning:

• \( A + B = B + A \)
• \( (A + B) + C = A + (B + C) \)

This allows flexibility when working with multiple matrices.

Matrix Subtraction

Matrix subtraction is similar to matrix addition, but instead of adding, you subtract corresponding elements. The condition for subtracting matrices is the same—they need to have the same dimensions.

Here's how you perform matrix subtraction:

• Take each element of the first matrix and subtract the corresponding element of the second matrix.
• Continue this process for all elements of the matrices involved.

Let's consider matrices \[A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\]The subtraction results in:\[A - B = \begin{bmatrix} a_{11}-b_{11} & a_{12}-b_{12} \ a_{21}-b_{21} & a_{22}-b_{22} \end{bmatrix}\]As in addition, order matters in subtraction, meaning \( A - B eq B - A \) unless all corresponding elements are equal, resulting in a matrix of all zeros.

Elementary Matrices

Elementary matrices are a special type of matrix used in various matrix operations, particularly to perform row operations. They play an essential role in algorithms for finding solutions to systems of linear equations, like Gaussian elimination.

Each elementary matrix is obtained by performing a single elementary row operation on the identity matrix. These operations include:

• Swapping two rows
• Multiplying a row by a nonzero scalar
• Adding a multiple of one row to another row

Consider the identity matrix \[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]If we swap the two rows, the corresponding elementary matrix becomes:\[E_1 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\]Applying these matrices to another matrix often helps in simplifying it or finding its inverse. Elementary matrices are key to understanding deeper linear algebra concepts, such as the theory behind matrix inversions and determinants.