Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression. $$ 4\left(\left[\begin{array}{rrr} -4 & 0 & 1 \\ 0 & 2 & 3 \end{array}\right]-\left[\begin{array}{rrr} 2 & 1 & -2 \\ 3 & -6 & 0 \end{array}\right]\right) $$

Step-by-Step Solution

Answer

The evaluated expression is \[ \left[ \begin{array}{ccc} -24 & -4 & 12 \ -12 & 32 & 12 \end{array} \right] \].

Step 1: Subtraction of Matrices

Subtract the two matrices elementwise. This means to subtract each element in the second matrix from the corresponding element in the first matrix. This will give: \[ \left[ \begin{array}{ccc} (-4-2) & (0-1) & (1 - -2) \ (0-3) & (2 - -6) & (3 - 0) \end{array} \right] = \left[ \begin{array}{ccc} -6 & -1 & 3 \ -3 & 8 & 3 \end{array} \right] \].

Step 2: Scalar Multiplication

Multiply each element in the resulting matrix by the scalar which is 4. This will give: \[ 4 \times -6 , 4 \times -1 , 4 \times 3 , 4 \times -3 , 4 \times 8 , 4 \times 3 = \left[ \begin{array}{ccc} -24 & -4 & 12 \ -12 & 32 & 12 \end{array} \right] \].

Key Concepts

Matrix Subtraction

Matrix subtraction is a straightforward yet essential operation in linear algebra. It involves taking two matrices of the same dimensions and subtracting the corresponding elements of one matrix from another. In order to perform matrix subtraction, each entry, or element, in the first matrix is decreased by the corresponding entry from the second matrix.
Let's consider a practical example to understand this better. If we have two matrices, Matrix A and Matrix B, then the subtraction of A and B is denoted as A - B. This operation yields a new matrix, where every element \( c_{ij} \) is the result of \( a_{ij} - b_{ij} \).
For our exercise, the matrices were subtracted as follows:

• The element in the first row, first column of the resulting matrix is \( -4 - 2 = -6 \).
• The element in the first row, second column is \( 0 - 1 = -1 \).
• This pattern continues for each corresponding pair of elements in the matrices.

Correctly performing this operation requires careful attention to ensure each element is accurately subtracted from its counterpart. An error in just one element will result in the entire matrix being incorrect.

Scalar Multiplication of Matrices

Once you understand matrix subtraction, the next concept to tackle is scalar multiplication of matrices. Scalar multiplication involves multiplying every element of a matrix by a constant value, known as a scalar. This operation is used frequently in scaling transformations and in mathematical equations relating to physical laws, among many other applications.
To carry out scalar multiplication, simply multiply every single element within the matrix by the scalar. If we represent the scalar by k and the matrix by A, then the scalar multiplication of matrix A by k is denoted as kA.

Applying Scalar Multiplication to the Example

After we subtracted the matrices in our exercise, we obtained a new matrix. To perform scalar multiplication, we then multiplied each of its elements by 4:

• \( 4 \times -6 = -24 \)
• \( 4 \times -1 = -4 \)
• And so on for the rest of the elements.

As a result of the scalar multiplication, every element in the matrix became larger or smaller by a factor of four, depending on the sign of the element.

Elementwise Matrix Operations

Elementwise matrix operations are performed by applying a specific arithmetic operation to each corresponding pair of elements from two matrices. It is crucial to note that for elementwise operations to be valid, the two matrices involved must have the same dimensions. In this case, operations such as addition, subtraction, multiplication, and division are conducted between corresponding elements of each matrix.
For example, in elementwise multiplication, each entry \( c_{ij} \) in the resulting matrix C is the product of entries \( a_{ij} \) and \( b_{ij} \) from matrices A and B, respectively. Similarly, elementwise subtraction was performed in the exercise to find the difference between two matrices before applying scalar multiplication.
Elementwise operations are important as they allow for the manipulation and analysis of matrices in various fields such as computer graphics, statistical analysis, and engineering. Correct application of these operations is fundamental to working with matrices and can lead to meaningful insights when interpreting the data they represent.