Problem 29 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The result of \(100D - 10E\) is \(\begin{bmatrix} -840 & 650 & -530 \\ 330 & 360 & 250 \\ -10 & 900 & 110 \end{bmatrix}\).

Step 1: Understand Matrix Dimensions

Matrix \(D\) is a \(3 \times 3\) matrix and Matrix \(E\) is also a \(3 \times 3\) matrix. Since both matrices have the same dimensions, they can be added or subtracted from each other after scaling, which is necessary for performing \(100D - 10E\) operation.

Step 2: Scale Matrix D by 100

Multiply each element of matrix \(D\) by 100. This results in:\[100D = \begin{bmatrix} -800 & 700 & -500 \ 400 & 300 & 200 \ 0 & 900 & 200 \end{bmatrix}\]

Step 3: Scale Matrix E by 10

Multiply each element of matrix \(E\) by 10. This results in:\[10E = \begin{bmatrix} 40 & 50 & 30 \ 70 & -60 & -50 \ 10 & 0 & 90 \end{bmatrix}\]

Step 4: Subtract Matrices

Subtract corresponding elements in \(10E\) from \(100D\):- First row: \([-800] - [40] = -840\), \([700] - [50] = 650\), \([-500] - [30] = -530\)- Second row: \([400] - [70] = 330\), \([300] - [-60] = 360\), \([200] - [-50] = 250\)- Third row: \([0] - [10] = -10\), \([900] - [0] = 900\), \([200] - [90] = 110\)This gives the resulting matrix:\[100D - 10E = \begin{bmatrix} -840 & 650 & -530 \ 330 & 360 & 250 \ -10 & 900 & 110 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is an operation where you add corresponding elements of two matrices of the same dimensions. This means the matrices must have the same number of rows and columns. If they do, you simply add each element in the first matrix to the element in the same position in the second matrix. For instance, if you have a 2x2 matrix A and another 2x2 matrix B, the resulting matrix after addition will also be 2x2.

• Example: If \(A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}\), then \(A + B = \begin{bmatrix} 1+5 & 2+6 \ 3+7 & 4+8 \end{bmatrix}\).
• Result: \(A + B = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}\).

It's important to remember that you cannot add matrices of different dimensions because there would be elements with no corresponding partners.

Matrix Subtraction

Matrix subtraction operates similarly to matrix addition. You subtract the elements of one matrix from the corresponding elements of another matrix, provided both matrices have the same dimensions. Each element in the result is found by subtracting an element in one matrix from the same-position element in the other matrix.

• Example: If \(A = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\), then \(A - B = \begin{bmatrix} 6-1 & 8-2 \ 10-3 & 12-4 \end{bmatrix}\).
• Result: \(A - B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}\).

Subtraction, just like addition, requires matrices to be of identical sizes. This ensures that each element in one matrix has a corresponding element in the other matrix to be subtracted from.

Matrix Scaling

Matrix scaling, or scalar multiplication, involves multiplying every element of a matrix by a constant value, known as the scalar. This operation does not change the dimensions of the matrix; it only alters its values. Scaling can be very useful when you need to adjust the magnitude of the matrix elements proportionally.

• Example: If you multiply matrix \(C = \begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}\) by a scalar 3, the result would be: \(3C = \begin{bmatrix} 6 & 9 \ 12 & 15 \end{bmatrix}\).
• Note: Each number inside the matrix is multiplied by the scalar 3.

This operation is foundational to methods like matrix subtraction after scaling, as demonstrated in the exercise with matrices \(D\) and \(E\), where each element of \(D\) and \(E\) is scaled by different scalars before being subtracted.

Matrix Dimensions

Understanding matrix dimensions is crucial for performing matrix operations. The dimensions of a matrix are determined by the number of rows and columns it contains, expressed as 'rows \(\times\) columns'. For instance, a matrix with 3 rows and 2 columns is called a 3x2 matrix.

• Example: \(\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\) is a 2x2 matrix because it has 2 rows and 2 columns.
• Size compatibility: To add or subtract matrices, they must have the same dimensions.
• Multiplication flexibility: For multiplication, however, the number of columns in the first matrix must match the number of rows in the second matrix.

In the original exercise, both matrices \(D\) and \(E\) were 3x3, hence allowing the scaling and subsequent subtraction operations. Knowing the dimensions helps to determine the eligibility and method of matrix operations.