Problem 16 - College Algebra with Modeling and Visualization

Question

( Refer to Examples 3-5.) LetA be the given matrix. Find \(A^{-1}\) without a calculator. $$ \left[\begin{array}{rr} 1 & 0 \\ 1 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse matrix \(A^{-1}\) is \(\begin{bmatrix} 1 & 0 \\ 1 & -1 \end{bmatrix}\).

Step 1: Identify the Matrix

The given matrix is \[A = \begin{bmatrix}1 & 0 \1 & -1\end{bmatrix}\] This is a 2x2 matrix, which is important because we will be using the formula for the inverse of a 2x2 matrix.

Step 2: Recall the Inverse Formula for 2x2 Matrix

For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the inverse is given by\[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]provided that \(ad - bc eq 0\). This value \(ad - bc\) is called the determinant of the matrix.

Step 3: Calculate the Determinant

Identify the elements in the matrix: \(a = 1\), \(b = 0\), \(c = 1\), and \(d = -1\).The determinant is calculated as \(ad - bc\):\[1(-1) - 0(1) = -1.\]Since the determinant is \(-1\), which is not zero, the inverse exists.

Step 4: Write the Expression for the Inverse

Using the inverse formula for a 2x2 matrix, substitute the values into the formula:\[A^{-1} = \frac{1}{-1} \begin{bmatrix} -1 & 0 \ -1 & 1 \end{bmatrix}\]This simplifies to:\[A^{-1} = -1 \times \begin{bmatrix} -1 & 0 \ -1 & 1 \end{bmatrix}\]

Step 5: Multiply to Find the Inverse

Multiply each entry of the matrix by \(-1\):\[A^{-1} = \begin{bmatrix} 1 & 0 \ 1 & -1 \end{bmatrix}\] Check the simplified result to confirm correctness.

Key Concepts

Determinant of a Matrix

When working with matrices, particularly in finding the inverse, understanding the determinant is crucial. The determinant is a special number that can tell us if a matrix is invertible. For a 2x2 matrix, the determinant is calculated using a simple formula. Given a matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is:\[ad - bc\]For example, consider the matrix \(\begin{bmatrix} 1 & 0 \ 1 & -1 \end{bmatrix}\). Let's identify its elements:

• \(a = 1\)
• \(b = 0\)
• \(c = 1\)
• \(d = -1\)

Plug these into the formula: \(1(-1) - 0(1) = -1\).The determinant is \(-1\). Since it is not zero, this matrix has an inverse. Always remember, a matrix is invertible if and only if its determinant is not zero.

Inverse of 2x2 Matrix

Finding the inverse of a 2x2 matrix is a straightforward process if you know the formula and the determinant of the matrix is not zero. The formula for the inverse of a 2x2 matrix is \[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]Using the earlier example, we have matrix \(\begin{bmatrix} 1 & 0 \ 1 & -1 \end{bmatrix}\) with a determinant of \(-1\). We substitute the elements and the determinant into the inverse formula:\[A^{-1} = \frac{1}{-1} \begin{bmatrix} -1 & 0 \ -1 & 1 \end{bmatrix}\]This can be simplified by multiplying each element inside the bracket by \(-1\), which gives us:\[A^{-1} = \begin{bmatrix} 1 & 0 \ 1 & -1 \end{bmatrix}\]Understanding how to use this formula will allow you to invert any 2x2 matrix confidently.

Matrix Algebra

Matrix algebra involves operations on matrices, similar to how Algebra deals with numbers and variables. It is a cornerstone of linear algebra, a subject that is key in various applications like computer graphics, physics, and engineering. Some basic matrix operations include:

• Matrix addition and subtraction: Adding or subtracting matrices involves adding or subtracting their corresponding elements.
• Scalar multiplication: Multiplying each element of the matrix by a scalar (a number).
• Matrix multiplication: Combining matrices in a way similar to multiplying numbers, but with special rules that involve multiplying rows by columns.
• Finding inverses: Used especially in solving system of linear equations, where the inverse matrix can help in finding the solution.

Understanding these fundamental matrix operations allows you to manipulate matrices efficiently and apply them to solve complex problems. Remember, practice makes perfect!