Problem 29 - College Algebra

Question

Use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{rr}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(100 D-10 E\)

Step-by-Step Solution

Answer

The result of \( 100D - 10E \) is \( \left[\begin{array}{rrr}-840 & 650 & -530 \\ 330 & 360 & 250 \\ -10 & 900 & 110\end{array}\right] \).

Step 1: Multiply Matrix D by 100

Matrix D is \( \left[\begin{array}{rrr}-8 & 7 & -5 \ 4 & 3 & 2 \ 0 & 9 & 2\end{array}\right] \). We will multiply every element of Matrix D by 100. This gives us: \[100 \times D = \left[\begin{array}{rrr}-800 & 700 & -500 \ 400 & 300 & 200 \ 0 & 900 & 200\end{array}\right]\]

Step 2: Multiply Matrix E by 10

Matrix E is \( \left[\begin{array}{rrr}4 & 5 & 3 \ 7 & -6 & -5 \ 1 & 0 & 9\end{array}\right] \). We will multiply every element of Matrix E by 10. This gives us: \[10 \times E = \left[\begin{array}{rrr}40 & 50 & 30 \ 70 & -60 & -50 \ 10 & 0 & 90\end{array}\right] \]

Step 3: Subtract 10E from 100D

We need to perform the subtraction \( 100D - 10E \) using the matrices from Steps 1 and 2.\[100D - 10E = \left[\begin{array}{rrr}-800 & 700 & -500 \ 400 & 300 & 200 \ 0 & 900 & 200\end{array}\right] - \left[\begin{array}{rrr}40 & 50 & 30 \ 70 & -60 & -50 \ 10 & 0 & 90\end{array}\right]\]Subtract each corresponding element:- First row: \(-800 - 40 = -840\), \(700 - 50 = 650\), \(-500 - 30 = -530\)- Second row: \(400 - 70 = 330\), \(300 - (-60) = 360\), \(200 - (-50) = 250\)- Third row: \(0 - 10 = -10\), \(900 - 0 = 900\), \(200 - 90 = 110\)Thus, \[100D - 10E = \left[\begin{array}{rrr}-840 & 650 & -530 \ 330 & 360 & 250 \ -10 & 900 & 110\end{array}\right]\]

Step 4: Final Result

The resulting matrix from the operation \( 100D - 10E \) is \[\left[\begin{array}{rrr}-840 & 650 & -530 \ 330 & 360 & 250 \ -10 & 900 & 110\end{array}\right]\].

Key Concepts

Matrix Subtraction

Matrix subtraction is a fundamental concept in matrix operations. It involves taking two matrices of the same dimensions and subtracting their corresponding elements.
For matrices to be subtracted, they must have an equal number of rows and columns. This condition ensures that each element in one matrix has a corresponding element in the other.
To perform matrix subtraction:

• Take each pair of corresponding elements from the two matrices.
• Subtract the element in the second matrix from the element in the first matrix.

As seen in the example with the matrices resulting from multiplying matrices D and E, subtraction is done element by element to create a new matrix. If matrices have different sizes, matrix subtraction cannot be carried out, showing the importance of matching dimensions.

Scalar Multiplication of Matrices

Scalar multiplication involves multiplying every element of a matrix by a single number, known as a scalar. This operation scales the entire matrix by a constant factor.
The process of scalar multiplication can be summarized as:

• Take the scalar value.
• Multiply each element of the matrix by the scalar.

In the exercise, to compute \(100D\) and \(10E\), each element in matrix D was multiplied by 100, and each element in matrix E was multiplied by 10. This operation is straightforward and involves only basic arithmetic, making it one of the more accessible types of matrix operations.

Matrix Arithmetic

Matrix arithmetic includes operations such as addition, subtraction, and multiplication. These operations are essential for various applications in mathematics and related fields.
When handling matrix arithmetic:

• Ensure matrices are compatible for operations by matching their dimensions.
• Follow the rules of matrix operations consistently for accurate results.
• Understand the properties of different operations. For instance, matrix addition and subtraction require matrices of the same size.

In this exercise, understanding scalar multiplication allowed for initial scaling of matrices D and E. Subsequent subtraction involved direct use of these modified matrices, demonstrating the power of combining basic operations for more complex arithmetic tasks in linear algebra.