Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Refer to the following matrices: \(A=\left[\begin{array}{rr}-1 & 2 \\ 3 & -2 \\ 4 & 0\end{array}\right] \quad B=\left[\begin{array}{rr}2 & 4 \\ 3 & 1 \\ -2 & 2\end{array}\right]\) \(C=\left[\begin{array}{rrr}3 & -1 & 0 \\ 2 & -2 & 3 \\ 4 & 6 & 2\end{array}\right] \quad D=\left[\begin{array}{rrr}2 & -2 & 4 \\ 3 & 6 & 2 \\\ -2 & 3 & 1\end{array}\right]\) Compute \(4 D-2 C\).

Step-by-Step Solution

Answer

The result of \(4D - 2C\) is: \(\left[\begin{array}{rrr}14 & -10 & 16 \\\ 16 & 20 & 14 \\\ 0 & 24 & 8\end{array}\right]\).

Step 1: Multiply the matrices with the scalars

First, we will multiply matrix \(C\) by the scalar \(-2\) and matrix \(D\) by the scalar \(4\). For matrix \(C\): \(-2 \times \left[\begin{array}{rrr}3 & -1 & 0 \\\ 2 & -2 & 3 \\\ 4 & 6 & 2\end{array}\right] = \left[\begin{array}{rrr}-6 & 2 & 0 \\\ -4 & 4 & -6 \\\ -8 & -12 & -4\end{array}\right]\) For matrix \(D\): \(4 \times \left[\begin{array}{rrr}2 & -2 & 4 \\\ 3 & 6 & 2 \\\ -2 & 3 & 1\end{array}\right] = \left[\begin{array}{rrr}8 & -8 & 16 \\\ 12 & 24 & 8 \\\ -8 & 12 & 4\end{array}\right]\)

Step 2: Subtract the matrices

Now, subtract the matrix we obtained by multiplying \(-2C\) from the matrix we obtained by multiplying \(4D\). \(\left[\begin{array}{rrr}8 & -8 & 16 \\\ 12 & 24 & 8 \\\ -8 & 12 & 4\end{array}\right] - \left[\begin{array}{rrr}-6 & 2 & 0 \\\ -4 & 4 & -6 \\\ -8 & -12 & -4\end{array}\right] = \left[\begin{array}{rrr}8-(-6) & -8-2 & 16-0 \\\ 12-(-4) & 24-4 & 8-(-6) \\\ -8-(-8) & 12-(-12) & 4-(-4)\end{array}\right]\)

Step 3: Calculate the result

Now, perform the subtraction within the matrix. \(\left[\begin{array}{rrr}14 & -10 & 16 \\\ 16 & 20 & 14 \\\ 0 & 24 & 8\end{array}\right]\) So, the result of \(4D - 2C\) is: \(\left[\begin{array}{rrr}14 & -10 & 16 \\\ 16 & 20 & 14 \\\ 0 & 24 & 8\end{array}\right]\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a mathematical operation that takes two matrices and produces a third matrix. It is different from simple arithmetic multiplication and requires specific rules to be followed. When you multiply two matrices, you perform a series of dot products between the rows of the first matrix and the columns of the second matrix. This operation is only possible when the number of columns in the first matrix matches the number of rows in the second matrix.
To perform matrix multiplication, it's key to remember a few things:

• The resulting matrix's size will be determined by the number of rows from the first matrix and the number of columns from the second matrix.
• If you have matrix A (with dimensions m×n) and matrix B (with dimensions n×p), the resulting matrix AB will have dimensions m×p.

It's important to note that matrix multiplication is not commutative, meaning that AB is not necessarily equal to BA. This aspect makes matrix multiplication unique compared to regular number multiplication.

Scalar Multiplication

Scalar multiplication involves multiplying each entry of a matrix by a constant value called a scalar. This operation is quite straightforward and maintains the dimensionality of the matrix. The result is a new matrix where each element is the product of the corresponding element in the original matrix and the scalar.
Here's how scalar multiplication works:

• Given a scalar, say 3, and a matrix, you multiply each element of the matrix by 3.
• If matrix A is \[\begin{array}{cc}1 & 2 \3 & 4 \\end{array}\], multiplying it by 3 gives you \[\begin{array}{cc}3 & 6 \9 & 12 \\end{array}\].

The operation is simple because it repeats the same multiplication across all elements. Scalar multiplication allows us to manipulate the scale of matrices without changing the structure or the position of elements within them.

Matrix Subtraction

Matrix subtraction is an operation that involves taking two matrices of the same dimensions and subtracting corresponding elements. The result is another matrix of the same size. For the subtraction to be possible, the two matrices must have identical dimensions.
Here is how you perform matrix subtraction:

• Ensure both matrices have the same number of rows and columns.
• Subtract the corresponding elements of the second matrix from the first matrix.
• For matrices A and B:\[\text{If } A = \begin{array}{cc}2 & 3 \4 & 5\end{array}, \text{and } B = \begin{array}{cc}1 & 1 \2 & 2\end{array}, \\text{then } A - B = \begin{array}{cc}1 & 2 \2 & 3\end{array}.\]

The resulting matrix from subtraction contains elements obtained by subtracting each component of the second matrix from the corresponding component of the first. This operation is straightforward and emphasizes on element-wise deduction.