Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Perform each operation if possible. $$2\left[\begin{array}{rr}2 & -1 \\ 5 & 1 \\ 0 & 3\end{array}\right]+\left[\begin{array}{rr}5 & 0 \\ 7 & -3 \\ 1 & 1\end{array}\right]-\left[\begin{array}{rr}9 & -4 \\ 4 & 4 \\ 1 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

The final result is \(\begin{bmatrix} 0 & 2 \\ 13 & -5 \\ 0 & 1 \end{bmatrix}\).

Step 1: Multiply the First Matrix by Scalar

Multiply each element in the first matrix by the scalar 2 as follows: \(2 \times \begin{bmatrix} 2 & -1 \ 5 & 1 \ 0 & 3 \end{bmatrix} = \begin{bmatrix} 2 \times 2 & 2 \times (-1) \ 2 \times 5 & 2 \times 1 \ 2 \times 0 & 2 \times 3 \end{bmatrix} = \begin{bmatrix} 4 & -2 \ 10 & 2 \ 0 & 6 \end{bmatrix}\)

Step 2: Add First Two Matrices

Add the resulting matrix from Step 1 to the second matrix element-wise:\(\begin{bmatrix} 4 & -2 \ 10 & 2 \ 0 & 6 \end{bmatrix} + \begin{bmatrix} 5 & 0 \ 7 & -3 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 4 + 5 & -2 + 0 \ 10 + 7 & 2 - 3 \ 0 + 1 & 6 + 1 \end{bmatrix} = \begin{bmatrix} 9 & -2 \ 17 & -1 \ 1 & 7 \end{bmatrix}\)

Step 3: Subtract the Third Matrix

Subtract the third matrix from the resulting matrix in Step 2, element-wise:\(\begin{bmatrix} 9 & -2 \ 17 & -1 \ 1 & 7 \end{bmatrix} - \begin{bmatrix} 9 & -4 \ 4 & 4 \ 1 & 6 \end{bmatrix} = \begin{bmatrix} 9 - 9 & -2 - (-4) \ 17 - 4 & -1 - 4 \ 1 - 1 & 7 - 6 \end{bmatrix} = \begin{bmatrix} 0 & 2 \ 13 & -5 \ 0 & 1 \end{bmatrix}\)

Step 4: Verify Result

Verify the final result by revisiting the addition and subtraction operations to ensure that all elements have been computed correctly without calculation errors.

Key Concepts

Scalar Multiplication

Scalar multiplication in matrix operations involves multiplying every element of a matrix by a single numerical value, known as the scalar. This process scales the entire matrix by that scalar value, which can either increase or decrease the values within the matrix based on whether the scalar is greater or less than one. In the original exercise, we are tasked with multiplying each element of the matrix \[\begin{bmatrix} 2 & -1 \ 5 & 1 \ 0 & 3 \end{bmatrix}\]by the scalar 2.

• First element: \(2 \times 2 = 4\)
• Second element: \(2 \times (-1) = -2\)
• Third element: \(2 \times 5 = 10\)
• Fourth element: \(2 \times 1 = 2\)
• Fifth element: \(2 \times 0 = 0\)
• Sixth element: \(2 \times 3 = 6\)

The resulting matrix becomes: \[\begin{bmatrix} 4 & -2 \ 10 & 2 \ 0 & 6 \end{bmatrix}\]This approach is straightforward, where each element is handled independently, reinforcing the simplicity of scalar multiplication in matrices.

Matrix Addition

Matrix addition is a fundamental operation where two matrices of the same dimensions are added together by combining their corresponding elements. To successfully perform matrix addition, both matrices must have the same number of rows and columns. In the exercise, after applying scalar multiplication, we then add the resulting matrix to another matrix \[\begin{bmatrix} 5 & 0 \ 7 & -3 \ 1 & 1 \end{bmatrix}\]element-wise. This means adding each element of the first matrix's corresponding position in the second matrix:

• First element: \(4 + 5 = 9\)
• Second element: \(-2 + 0 = -2\)
• Third element: \(10 + 7 = 17\)
• Fourth element: \(2 + (-3) = -1\)
• Fifth element: \(0 + 1 = 1\)
• Sixth element: \(6 + 1 = 7\)

The output of this operation is a new matrix:\[\begin{bmatrix} 9 & -2 \ 17 & -1 \ 1 & 7 \end{bmatrix}\]Matrix addition emphasizes the importance of matching the structure of matrices to ensure every element is appropriately aligned for addition.

Element-wise Operations

Element-wise operations refer to mathematical operations performed between matrices element by element. These operations can include addition, subtraction, multiplication, or division, but must be done between matrices of the same sizes.
In the context of our problem, both the addition and the subtraction steps are handled in an element-wise manner.
Each step involves processing corresponding elements at the same index position from each involved matrix.
For instance, in the addition and subtraction processes, each step selectively focuses on individual elements:

• Addition: Each element from the matrices is increased by its counterpart from the other matrix.
• Subtraction: Each element has its corresponding element subtracted to form the resultant matrix.

Element-wise operations are particularly useful for handling large datasets in parallel and simplify complex problem-solving by natural parallelism. The concept underscores precise alignment and matching dimensions as critical for success.

Matrix Subtraction

Matrix subtraction is performed similarly to matrix addition, by subtracting corresponding elements of the matrices involved. The requirement is that both matrices have the same dimensions, enabling each element in one matrix to be subtracted from the corresponding element in the other.
In the exercise, the matrix obtained from the addition step is \[\begin{bmatrix} 9 & -2 \ 17 & -1 \ 1 & 7 \end{bmatrix}\]and the matrix we subtract is\[\begin{bmatrix} 9 & -4 \ 4 & 4 \ 1 & 6 \end{bmatrix}\]Subtract each corresponding element:

• First element: \(9 - 9 = 0\)
• Second element: \(-2 - (-4) = 2\)
• Third element: \(17 - 4 = 13\)
• Fourth element: \(-1 - 4 = -5\)
• Fifth element: \(1 - 1 = 0\)
• Sixth element: \(7 - 6 = 1\)

The final resulting matrix is:\[\begin{bmatrix} 0 & 2 \ 13 & -5 \ 0 & 1 \end{bmatrix}\]This operation wraps up the sequence, providing a clear example of how Matrix Subtraction works under the rules of element-wise operations.