Problem 19 - Precalculus with Limits

Question

In Exercises 19-24, evaluate the expression. \(\left[ \begin{array}{r} -5 & 0 \\ 3 & -6 \end{array} \right] + \left[ \begin{array}{r} 7 & 1 \\ -2 & -1 \end{array} \right] + \left[ \begin{array}{r} -10 & -8 \\ 14 & 6 \end{array} \right]\)

Step-by-Step Solution

Answer

So the sum of the three given matrices is \(\left[\begin{array}{r} -8 & -7 \ 15 & -1 \end{array} \right]\)

Step 1: Add First Two Matrices

First, add the equivalent elements in the first two matrices: \(\left[ \begin{array}{r} -5 & 0 \ 3 & -6 \end{array} \right] + \left[\begin{array}{r} 7 & 1 \ -2 & -1 \end{array} \right] = \left[\begin{array}{r} -5+7 & 0+1 \ 3+(-2) & -6+(-1) \end{array} \right] = \left[\begin{array}{r} 2 & 1 \ 1 & -7 \end{array} \right]\)

Step 2: Adding Resulting Matrix to the Third Matrix

Next, add the resulting matrix from the step 1 to the third matrix: \(\left[\begin{array}{r} 2 & 1 \ 1 & -7 \end{array} \right] + \left[\begin{array}{r} -10 & -8 \ 14 & 6 \end{array} \right] = \left[\begin{array}{r} 2+(-10) & 1+(-8) \ 1+(14) & -7+6 \end{array} \right] = \left[\begin{array}{r} -8 & -7 \ 15 & -1 \end{array} \right]\)

Key Concepts

Element-Wise Addition

Matrix addition is a straightforward yet important operation in linear algebra. When performing element-wise addition, each element in one matrix is added to the corresponding element in another matrix. Matrices must be of the same dimensions to add them together.
For example, if you have two matrices:

• Matrix A: \( \begin{bmatrix} -5 & 0 \ 3 & -6 \end{bmatrix} \)
• Matrix B: \( \begin{bmatrix} 7 & 1 \ -2 & -1 \end{bmatrix} \)

You add the elements from Matrix A to Matrix B "element-wise" like this:

• The first row, first column: \(-5 + 7 = 2\)
• The first row, second column: \(0 + 1 = 1\)
• The second row, first column: \(3 + (-2) = 1\)
• The second row, second column: \(-6 + (-1) = -7\)

The result is a new matrix: \( \begin{bmatrix} 2 & 1 \ 1 & -7 \end{bmatrix} \). This process is repeated when adding with a third matrix.

Matrix Operations

Matrix operations involve manipulating matrices to perform various algebraic tasks such as addition, subtraction, or multiplication. When dealing with multiple matrices in operations, the rules are simple but strict:

• Matrices must be of the same size to be added or subtracted.
• Matrix multiplication requires the number of columns in the first matrix to match the number of rows in the second.

In the given exercise, the primary focus is on addition. Here, the step-by-step solution reveals the process of adding matrices sequentially, ensuring each pair of matrices is compatible in size. This highlights the need for methodical execution in operations to find accurate results.
It’s essential to remember that unlike simple numbers, matrices involve multi-dimensional interplay, demanding both precision and understanding of underlying rules.

Matrices Evaluation

Evaluating matrices refers to the process of assessing their operations and determining the resultant matrix from tasks such as addition. The exercise you encountered involves successfully evaluating the sum of three matrices.
The steps provided ensured:

• Compatibility of matrices for operation through matching dimensions.
• Careful, step-by-step processing of each element pair in the matrices.
• Determining a final matrix that accurately reflects the summation.

The final matrix obtained was:\[\begin{bmatrix} -8 & -7 \ 15 & -1 \end{bmatrix}\]This outcome results from first adding the elements of the first two matrices, forming an intermediate matrix, and then adding the elements of this matrix to the third matrix. Evaluating matrices in this structured way ensures both the reliability and accuracy of results. It reflects the beauty of matrix algebra through clarity and systematic approach.