Problem 65 - College Algebra

Question

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$\left[\begin{array}{rr}3 & -1 \\\\-2 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

The multiplicative inverse of the matrix \( \left[\begin{array}{rr}3 & -1 \\-2 & 1\end{array}\right] \) is \( \left[\begin{array}{rr}1 & 1 \\-2 & 3\end{array}\right] \).

Step 1: Calculate the inverse using a graphing utility

Enter the matrix \( \left[\begin{array}{rr}3 & -1 \\-2 & 1\end{array}\right] \) into your graphing utility and use the appropriate function to calculate the inverse. The result should be a 2x2 matrix.

Step 2: Write down the calculated inverse

The graphing utility should provide an inverse matrix of \( \left[\begin{array}{rr}1 & 1 \\-2 & 3\end{array}\right] \). This is the multiplicative inverse of our original matrix.

Step 3: Verify the inverse

To confirm that the inverse matrix is accurate, you want to multiply the original matrix by the calculated inverse, in that order. If the inverse is correct, the result should be the 2x2 identity matrix, \( \left[\begin{array}{rr}1 & 0 \\-0 & 1\end{array}\right] \). Multiply the original matrix by the inverse to confirm.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools often used in mathematics to visually display equations, solve problems, and handle complex calculations. These utilities can be software programs on computers or apps on graphing calculators. For matrix operations, they provide functions to input matrices and perform calculations such as finding the inverse.
When you enter a matrix into a graphing utility, it simplifies the process by removing the need for manual calculations that can be time-consuming and prone to human error. Using these tools can ensure precision, especially with cumbersome calculations like those involving matrix manipulation.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, essential for verifying the correctness of matrix inverses. It involves the calculation of the dot product of the rows of the first matrix with the columns of the second matrix.
To multiply two matrices, follow these steps:

• Ensure the number of columns in the first matrix matches the number of rows in the second matrix.
• Multiply each element of a row in the first matrix by the corresponding element of a column in the second matrix.
• Sum these products to get a single element in the result matrix.

The accuracy of matrix inverses is verified through this multiplication process. If an original matrix and its inverse yield an identity matrix upon multiplication, the inverse is confirmed as correct.

Identity Matrix

An identity matrix is a special kind of matrix in which all elements of the principal diagonal are ones and all other elements are zeros. For any matrix A, the product of A and an identity matrix of compatible size results in A itself. This property works much like multiplying a number by one.
Identity matrices are crucial in verifying matrix inverses. In our context of 2x2 matrices, the identity matrix is \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). If multiplying a matrix by its inverse yields the identity matrix, it confirms that the inverse is correct. This is why identity matrices play a vital role in linear algebra, serving as the benchmark for accuracy in matrix inversion.

2x2 Matrices

2x2 matrices are among the simplest forms of matrices used in linear algebra. These matrices consist of two rows and two columns, and they form the basis for understanding larger matrices and more complex matrix operations.
The inverse of a 2x2 matrix, if it exists, can be calculated using the formula:\[A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]where \(a, b, c,\) and \(d\) are the elements of the matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), and \(ad-bc eq 0\) (this is called the determinant of the matrix).
Understanding how to manipulate these matrices is essential for engaging with more complex concepts in linear algebra, including matrix inverses, systems of equations, and eigenvalues.