Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Inverse of a Matrix, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrr} 1 & 1 & 2 \\ 3 & 1 & 0 \\ -2 & 0 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The method involves inputting the given matrix into a graphing utility and using the built-in function to calculate the inverse.

Step 1: Analyzing the Matrix

Before finding the inverse, first identify the given matrix. In this case, that's a 3x3 matrix with elements 1, 1, 2, in the first row, 3, 1, 0 in the second row, and -2, 0, 3 in the third row.

Step 2: Checking the Determinant

Remember, in order for a matrix to have an inverse, the determinant of the matrix has to be non-zero. Starting by calculating the determinant of the given matrix using formula will result: Det(A) = aei + bfg + cdh - ceg - bdi - afh. Here, a=1, b=1, c=2, d=3, e=1, f=0, g=-2, h=0, i=3. So, calculating gives the answer Det(A) = 1x1x3 + 1x0x(-2) + 2x3x0 - 2x1x(-2) - 1x0x3 - 1x3x1 = 3 + 4 - 3 = 4. Because the determinant is not zero, an inverse of this matrix exists.

Step 3: Input Matrix to Graphing Utility

To calculate the inverse of the matrix, input the original matrix into the graphing utility. While every graphing utility is different, look for the option to enter matrices and input the values exactly as they appear in the original matrix.

Step 4: Calculate Inverse Matrix

Using the graphing utility's function for finding the inverse of a matrix (often denoted as 'inv' or similar), perform the operation on the input matrix. The output matrix given by the software is the inverse of the original matrix.

Key Concepts

Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements. It provides crucial information about the matrix, such as whether it is invertible or not. This number is a scalar value and is only defined for square matrices.
The determinant tells us:

• If the determinant is zero, the matrix does not have an inverse.
• If the determinant is non-zero, the matrix has an inverse.

To find the determinant of a 3x3 matrix like the one in the exercise, we use a specific formula:\[ ext{Det}(A) = aei + bfg + cdh - ceg - bdi - afh\]Here, each letter corresponds to an element of the matrix, positioned in the rows and columns of the matrix. Calculating the determinant accurately is critical as it determines the possibility of an inverse existing.

Graphing Utility

A graphing utility is a technological tool commonly used in mathematics to perform complex calculations, such as finding the inverse of a matrix or graphing functions. These utilities can be applications on calculators or software programs on computers.
For matrices, a graphing utility can:

• Store multiple matrices and their elements.
• Calculate determinants and inverses quickly.

Using a graphing utility often involves inputting the matrix exactly as presented, ensuring all values are accurate. Many graphing utilities have a dedicated function, often labeled 'inv', specifically for finding inverses of matrices.
These tools significantly reduce the manual effort needed for complex calculations and provide precise results when used correctly.

3x3 Matrix

A 3x3 matrix is a square grid with three rows and three columns, containing nine elements. In matrix algebra, these elements are central to various calculations like determinants and inverses.
For example, the matrix in the problem is:\[ \begin{bmatrix} 1 & 1 & 2 \ 3 & 1 & 0 \ -2 & 0 & 3 \end{bmatrix}\]This matrix has:

• Elements: Nine elements where each entry represents a distinct number or variable.
• Square Shape: The same number of rows and columns, which is necessary for it to potentially have an inverse.

A 3x3 matrix is significant in more advanced mathematics as it represents systems with three variables or transformations in three-dimensional space, making it a fundamental format for various problems.

Determinant Calculation

Calculating the determinant of a matrix involves using a set mathematical formula. This process might seem lengthy but becomes straightforward with practice.
For a 3x3 matrix, follow these steps:

• Identify the elements of the matrix, such as \(a, b, c, d, e, f, g, h,\) and \(i\).
• Substitute these elements into the determinant formula. \[ ext{Det}(A) = aei + bfg + cdh - ceg - bdi - afh \]
• Calculate each multiplication and addition carefully.

For instance, in the specific exercise provided, substituting the values:

• \(1 \cdot 1 \cdot 3 = 3\)
• \(1 \cdot 0 \cdot (-2) = 0\)
• \(2 \cdot 3 \cdot 0 = 0\)
• \(-2 \cdot 1 \cdot -2 = 4\)
• \(-1 \cdot 0 \cdot 3 = 0\)
• \(-1 \cdot 3 \cdot 1 = -3\)
• Add or subtract these results as the formula suggests to obtain \(4\).

Accurate determinant calculation confirms the potential of finding an inverse.