Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Perform each operation if possible. $$\left[\begin{array}{rrr}6 & -2 & 4 \\ -2 & 5 & 8 \\ 1 & 0 & 2\end{array}\right]+\left[\begin{array}{rrr}3 & 0 & 8 \\ 1 & -2 & 4 \\ 6 & 9 & -2\end{array}\right]-\left[\begin{array}{rrr}-4 & 2 & 1 \\ 0 & 3 & -2 \\ 4 & 2 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The result of the matrix operations is \( \begin{bmatrix} 13 & -4 & 11 \\ -1 & 0 & 14 \\ 3 & 7 & 0 \end{bmatrix} \).

Step 1: Validate Matrix Dimensions

Check if the matrices have the same dimensions to perform addition and subtraction. The given matrices are all 3x3 matrices, so it is possible to perform the operations.

Step 2: Perform Matrix Addition

Add the first two matrices element-wise. The sum of the matrices is:\[\begin{bmatrix}6 & -2 & 4 \-2 & 5 & 8 \1 & 0 & 2\end{bmatrix} + \begin{bmatrix}3 & 0 & 8 \1 & -2 & 4 \6 & 9 & -2\end{bmatrix} = \begin{bmatrix}9 & -2 & 12 \-1 & 3 & 12 \7 & 9 & 0\end{bmatrix}\]

Step 3: Perform Matrix Subtraction

Subtract the third matrix from the resulting matrix obtained in Step 2. The subtraction is performed element-wise:\[\begin{bmatrix}9 & -2 & 12 \-1 & 3 & 12 \7 & 9 & 0\end{bmatrix} - \begin{bmatrix}-4 & 2 & 1 \0 & 3 & -2 \4 & 2 & 0\end{bmatrix} = \begin{bmatrix}13 & -4 & 11 \-1 & 0 & 14 \3 & 7 & 0\end{bmatrix}\]

Step 4: Compile Final Result

The final result obtained after performing both the addition and subtraction operations is:\[\begin{bmatrix}13 & -4 & 11 \-1 & 0 & 14 \3 & 7 & 0\end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. It involves adding two matrices by adding corresponding elements together. The key requirement for this operation is that both matrices must have the same dimensions.

For example, to add two 3x3 matrices, you align them based on their structure and add each element from one matrix to the matching element in the other matrix.

Consider adding two matrices:

• Matrix A: \(\begin{bmatrix}6 & -2 & 4 \-2 & 5 & 8 \1 & 0 & 2\end{bmatrix}\)
• Matrix B:\(\begin{bmatrix}3 & 0 & 8 \1 & -2 & 4 \6 & 9 & -2\end{bmatrix} \)

You add them by summing corresponding entries:

• First row: \(6 + 3 = 9\), \(-2 + 0 = -2\), \(4 + 8 = 12\)
• Second row: \(-2 + 1 = -1\), \(5 + (-2)= 3\), \(8 + 4 = 12\)
• Third row: \(1 + 6 = 7\), \(0 + 9 = 9\), \(2 + (-2) = 0\)

The sum:\(\begin{bmatrix}9 & -2 & 12 \-1 & 3 & 12 \7 & 9 & 0\end{bmatrix}\)

Matrix Subtraction

Matrix subtraction works similarly to matrix addition, but instead of adding the elements of corresponding positions, you subtract them. Like addition, subtraction requires the matrices to have the same dimensions.

To subtract one matrix from another, align the matrices and subtract each entry in the same position from one another. Let's continue from our previous example where we need to subtract a third matrix, Matrix C, from the result we obtained:

• Matrix C:\(\begin{bmatrix}-4 & 2 & 1 \0 & 3 & -2 \4 & 2 & 0\end{bmatrix}\)

Perform element-wise subtraction from the previous addition result:

• First row: \(9 - (-4) = 13\), \(-2 - 2 = -4\), \(12 - 1 = 11\)
• Second row: \(-1 - 0 = -1\), \(3 - 3 = 0\), \(12 - (-2) = 14\)
• Third row: \(7 - 4 = 3\), \(9 - 2 = 7\), \(0 - 0 = 0\)

The difference:\(\begin{bmatrix}13 & -4 & 11 \-1 & 0 & 14 \3 & 7 & 0\end{bmatrix}\)

Matrix Dimensions

Matrix dimensions are crucial for determining whether matrix operations like addition and subtraction are possible. A matrix's dimensions are given as 'rows x columns'.

In the case of our example, we dealt with 3x3 matrices, indicating each has 3 rows and 3 columns. This uniformity ensured that the operations could proceed easily.

Here are some key points to remember:

• To add or subtract matrices, their dimensions must be identical. For instance, two matrices both need to be 2x3 or 3x3.
• If dimensions do not match, you cannot perform these element-wise operations.
• Knowing dimensions also helps in understanding the structure and extends to more complex operations like matrix multiplication.

Always make the dimension check the first step before commencing any matrix arithmetic operations to avoid errors.