Problem 29 - College Algebra

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. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {6} & {7}\end{array}\right], B=\left[\begin{array}{cc}{-9} & {6} \\ {-4} & {2}\end{array}\right], C=\left[\begin{array}{cc}{0} & {9} \\ {7} & {1}\end{array}\right], D=\left[\begin{array}{rrr}{-8} & {7} & {-5} \\ {4} & {3} & {2} \\ {0} & {9} & {2}\end{array}\right], E=\left[\begin{array}{rrr}{4} & {5} & {3} \\ {7} & {-6} & {-5} \\ {1} & {0} & {9}\end{array}\right] $$ $$ 100 D-10 E $$

Step-by-Step Solution

Answer

Result is \( \begin{bmatrix}-840 & 650 & -530 \\ 330 & 360 & 250 \\ -10 & 900 & 110 \end{bmatrix} \).

Step 1: Identify Matrix Dimensions

Matrix \( D \) is \( 3 \times 3 \) and matrix \( E \) is also \( 3 \times 3 \). Since both matrices have the same dimensions, they can be multiplied by scalar factors and subtracted from one another.

Step 2: Perform Scalar Multiplication on Matrix D

Multiply each element of matrix \( D \) by 100:\[100D = \begin{bmatrix}100 \times -8 & 100 \times 7 & 100 \times -5 \100 \times 4 & 100 \times 3 & 100 \times 2 \100 \times 0 & 100 \times 9 & 100 \times 2\end{bmatrix} = \begin{bmatrix}-800 & 700 & -500 \400 & 300 & 200 \0 & 900 & 200\end{bmatrix}\]

Step 3: Perform Scalar Multiplication on Matrix E

Multiply each element of matrix \( E \) by 10:\[10E = \begin{bmatrix}10 \times 4 & 10 \times 5 & 10 \times 3 \10 \times 7 & 10 \times -6 & 10 \times -5 \10 \times 1 & 10 \times 0 & 10 \times 9\end{bmatrix} = \begin{bmatrix}40 & 50 & 30 \70 & -60 & -50 \10 & 0 & 90\end{bmatrix}\]

Step 4: Subtract Matrices

Subtract the scaled matrix \( 10E \) from the scaled matrix \( 100D \):\[100D - 10E = \begin{bmatrix}-800 & 700 & -500 \400 & 300 & 200 \0 & 900 & 200\end{bmatrix} - \begin{bmatrix}40 & 50 & 30 \70 & -60 & -50 \10 & 0 & 90\end{bmatrix} = \begin{bmatrix}-840 & 650 & -530 \330 & 360 & 250 \-10 & 900 & 110\end{bmatrix}\]

Step 5: Conclusion

The result of the operation \( 100D - 10E \) is the matrix:\[ \begin{bmatrix}-840 & 650 & -530 \330 & 360 & 250 \-10 & 900 & 110\end{bmatrix} \] This is the final answer.

Key Concepts

Scalar Multiplication

Scalar multiplication in matrices is a straightforward yet powerful operation. It involves multiplying every single element of a matrix by a constant number, known as the scalar. This operation is akin to scaling the values of the matrix by the same factor.
For matrix operations, scalar multiplication is used frequently to adjust the scale of matrix components without changing the structure of the matrix itself.
For example, if we have a matrix \[D = \begin{bmatrix}-8 & 7 & -5 \ 4 & 3 & 2 \ 0 & 9 & 2\end{bmatrix}\]and we multiply it by a scalar 100, each element of the matrix D will be multiplied by 100.
So, \[100D = \begin{bmatrix}100 \times -8 & 100 \times 7 & 100 \times -5 \ 100 \times 4 & 100 \times 3 & 100 \times 2 \ 100 \times 0 & 100 \times 9 & 100 \times 2\end{bmatrix} = \begin{bmatrix}-800 & 700 & -500 \ 400 & 300 & 200 \ 0 & 900 & 200\end{bmatrix}\]
This operation is vital in adjusting the magnitude of matrices keeping the relative differences the same.

Matrix Subtraction

Matrix subtraction involves element-wise subtraction of corresponding elements in two matrices. This operation requires that the matrices have the same dimensions, meaning they must have the same number of rows and columns.
This ensures that each element in one matrix has a matching element in the other matrix to subtract from.
For example, given the matrices \[100D = \begin{bmatrix}-800 & 700 & -500 \ 400 & 300 & 200 \ 0 & 900 & 200\end{bmatrix}\] and\[10E = \begin{bmatrix}40 & 50 & 30 \ 70 & -60 & -50 \ 10 & 0 & 90\end{bmatrix}\], subtraction is performed as:\[100D - 10E = \begin{bmatrix}-800 & 700 & -500 \ 400 & 300 & 200 \ 0 & 900 & 200\end{bmatrix} - \begin{bmatrix}40 & 50 & 30 \ 70 & -60 & -50 \ 10 & 0 & 90\end{bmatrix} = \begin{bmatrix}-840 & 650 & -530 \ 330 & 360 & 250 \ -10 & 900 & 110\end{bmatrix}\]
This operation effectively blends these two matrices into one, each calculated by subtracting one element from its corresponding counterpart in the other matrix.

Matrix Dimensions

The dimensions of a matrix are defined by its number of rows and columns, written as "rows × columns". Understanding matrix dimensions is crucial in performing operations like addition, subtraction, and multiplication because these operations are only defined for matrices with compatible dimensions.
For addition and subtraction to be possible, two matrices must have exactly the same dimensions.
In our example, both matrices D and E are 3 x 3 matrices, meaning they have three rows and three columns each.
This alignment allows scalar multiplication on both, and subsequently, subtraction to produce a valid result matrix.
It's always essential to check dimensions first before attempting matrix operations to ensure the operations can be performed successfully.