Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations. \(0.5\left[\begin{array}{rrr}1 & 3 & 5 \\ 5 & 2 & -1 \\ -2 & 0 & 1\end{array}\right]-0.2\left[\begin{array}{rrr}2 & 3 & 4 \\ -1 & 1 & -4 \\ 3 & 5 & -5\end{array}\right]\) \(+0.6\left[\begin{array}{rrr}3 & 4 & -1 \\ 4 & 5 & 1 \\ 1 & 0 & 0\end{array}\right]\)

Step-by-Step Solution

Answer

The short answer for the given matrix operation is: \( \begin{bmatrix} 1.9 & 3.3 & 1.1 \\ 5.1 & 3.8 & 0.9 \\ -1 & -1 & 1.5 \end{bmatrix} \)

Step 1: Perform scalar multiplications of matrices with the given scalars

First we will multiply each matrix by its corresponding scalar: \( 0.5\begin{bmatrix} 1 & 3 & 5 \\ 5 & 2 & -1 \\ -2 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 0.5 & 1.5 & 2.5 \\ 2.5 & 1 & -0.5 \\ -1 & 0 & 0.5 \end{bmatrix} \) \( 0.2\begin{bmatrix} 2 & 3 & 4 \\ -1 & 1 & -4 \\ 3 & 5 & -5 \end{bmatrix} = \begin{bmatrix} 0.4 & 0.6 & 0.8 \\ -0.2 & 0.2 & -0.8 \\ 0.6 & 1 & -1 \end{bmatrix} \) \( 0.6\begin{bmatrix} 3 & 4 & -1 \\ 4 & 5 & 1 \\ 1 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 1.8 & 2.4 & -0.6 \\ 2.4 & 3 & 0.6 \\ 0.6 & 0 & 0 \end{bmatrix} \)

Step 2: Add the resulting matrices

Now, we will add the matrices obtained in step 1: \( \begin{bmatrix} 0.5 & 1.5 & 2.5 \\ 2.5 & 1 & -0.5 \\ -1 & 0 & 0.5 \end{bmatrix} - \begin{bmatrix} 0.4 & 0.6 & 0.8 \\ -0.2 & 0.2 & -0.8 \\ 0.6 & 1 & -1 \end{bmatrix} + \begin{bmatrix} 1.8 & 2.4 & -0.6 \\ 2.4 & 3 & 0.6 \\ 0.6 & 0 & 0 \end{bmatrix} \) Now, we add the corresponding elements of each matrix: \( \begin{bmatrix} (0.5-0.4+1.8) & (1.5-0.6+2.4) & (2.5-0.8-0.6) \\ (2.5+0.2+2.4) & (1-0.2+3) & (-0.5+0.8+0.6) \\ (-1-0.6+0.6) & (0-1+0) & (0.5+1+0) \end{bmatrix} \)

Step 3: Calculate the resulting matrix

Finally, we compute the resulting matrix by simplifying each element: \( \begin{bmatrix} (1.9) & (3.3) & (1.1) \\ (5.1) & (3.8) & (0.9) \\ (-1) & (-1) & (1.5) \end{bmatrix} \) So, the final result of the indicated operations is: \( \begin{bmatrix} 1.9 & 3.3 & 1.1 \\ 5.1 & 3.8 & 0.9 \\ -1 & -1 & 1.5 \end{bmatrix} \)

Key Concepts

Scalar Multiplication of Matrices

Scalar multiplication of matrices is a fundamental operation in linear algebra. It involves multiplying every entry of a matrix by a fixed number, known as a scalar. For instance, multiplying the matrix

\( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \)
by the scalar \( k \) results in

\( kA = \begin{bmatrix} ka & kb \ kc & kd \end{bmatrix} \).

The beauty of scalar multiplication lies in its simplicity and its role in stretching or shrinking a matrix geometrically. When a matrix representing certain data or transformations is scaled, each of its components is uniformly affected, just like magnifying or reducing an image's size while keeping its proportions intact. In the exercise provided, scalar multiplication is the initial step in manipulating the matrices, setting the stage for further operations.

Matrix Addition

Matrix addition is another essential procedure in matrix operations, where two matrices of identical dimensions are added together by summing their corresponding entries. This operation is performed element-wise, which means we add the numbers that reside in the same position in each of the matrices. For example, consider adding two 2x2 matrices:

\( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) and \( B = \begin{bmatrix} e & f \ g & h \end{bmatrix} \),

the sum, \( A + B \), would be

\( \begin{bmatrix} a+e & b+f \ c+g & d+h \end{bmatrix} \).

This operation is straightforward yet powerful, as it's used to combine transformations or datasets represented by matrices. In the given exercise, after scalar multiplication, matrix addition is adeptly used to combine the scaled matrices into a single new matrix.

Elementary Matrix Arithmetic

Elementary matrix arithmetic includes operations such as addition, subtraction, and scalar multiplication.

Understanding Matrix Compatibility

For operations like addition and subtraction, matrices must be compatible—they should have the same dimensions, meaning the same number of rows and columns. If this condition is met, operations are conducted element by element.

Commutativity and Associativity

Importantly, matrix addition is both commutative and associative. The commutative property means that \( A + B = B + A \), and the associative property means that \( (A + B) + C = A + (B + C) \). However, matrix multiplication (not covered in this exercise) does not generally obey the commutative law.

Subtraction and Scalar Multiplication

Subtraction follows the same rules as addition but works with differences. Scalar multiplication, as previously described, affects every element of a matrix individually. These operations are the building blocks of more complex matrix manipulations used extensively in various applications such as solving linear equations, graphics transformations, and more.