Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{rr} 1 & 2 \\ -2 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of A is 4, while the determinant of its inverse is 0.25. These two are reciprocals of each other, leading to the conjecture that the determinant of a matrix multiplied by the determinant of its inverse always equals 1.

Step 1: Find the determinant of A

The determinant of a 2x2 matrix \[ A = \left[\begin{array}{cc} a & b \\ c & d \end{array}\right] \] is calculated as \( \operatorname{det}(A) = ad - bc \). So, for our matrix A = \left[\begin{array}{rr} 1 & 2 \\ -2 & 2 \end{array}\right], the determinant is (1*2) - (-2*2) which equals 4.

Step 2: Find the inverse of A

The formula for the inverse of a 2x2 matrix is given by \[ A^{-1} = \frac{1}{\operatorname{det}(A)} \left[\begin{array}{rr} d & -b \\ -c & a \end{array}\right] \] In this case, A = \left[\begin{array}{rr} 1 & 2 \\ -2 & 2 \end{array}\right] has a determinant of 4, b=2, c=-2, a=1 and d=2. Substituting these values into the formula gives \[ A^{-1} = \left[\begin{array}{cc} \frac{2}{4} & \frac{-2}{4} \\ \frac{2}{4} & \frac{1}{4} \end{array}\right] = \left[\begin{array}{cc} 0.5 & -0.5 \\ 0.5 & 0.25 \end{array}\right] \]

Step 3: Find the determinant of \(A^{-1}\)

Using the same determinant calculation process as above, we can find the determinant of \(A^{-1}\) by plugging its values into the formula ad-bc = (0.5*0.25) - (-0.5*0.5). That equals 0.25.

Step 4: Compare the results and make a conjecture

Comparing our results, we saw that the determinant of A was 4 and the determinant of its inverse was 0.25. These two numbers are reciprocals of each other. So, we can conjecture that for any 2x2 matrix \(A\), \(\operatorname{det}(A) \times \operatorname{det}(A^{-1}) = 1\).

Key Concepts

Matrix Inverse

Understanding the concept of a matrix inverse is essential for various calculations in linear algebra. Specifically, the inverse of a matrix, denoted as \(A^{-1}\), is a matrix that, when multiplied by the original matrix \(A\), gives the identity matrix. The identity matrix is a special kind of matrix that acts like the number 1 for matrix multiplication. This means that \(A \times A^{-1} = I\), where \(I\) is the identity matrix. For a 2x2 matrix, finding the inverse involves a specific formula:

\[ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \bs{bmatrix} \].

Remember, a matrix must be square (same number of rows and columns) to have an inverse, and not all square matrices have inverses. Matrices without an inverse are called singular or non-invertible. A matrix will have an inverse only if its determinant is non-zero. The determinant is a scalar value that summarizes certain properties of the matrix.

2x2 Matrix

Matrix operations are fundamental concepts in linear algebra, and understanding the operations for a 2x2 matrix can be a great introduction to this topic. A 2x2 matrix is a square matrix with two rows and two columns, typically represented as:

\[ \begin{bmatrix} a & b \ c & d \ \bs{bmatrix} \].

The determinant for this structure takes a formula succinctly expressed as \(\text{det}(A) = ad - bc\). The value obtained from this formula plays a significant role in various applications, including calculating the inverse, solving systems of linear equations, and describing geometric transformations. It is important to note that simplicity of the 2x2 structure makes it easier to grasp these foundational concepts and apply them to more complex matrices.

Reciprocal Property of Determinants

The reciprocal property of determinants is a fascinating feature observed in linear algebra when dealing with square matrices and their inverses. It posits that the determinant of a matrix multiplied by the determinant of its inverse yields the number 1. This can be mathematically expressed as:

\( \text{det}(A) \times \text{det}(A^{-1}) = 1 \).

In the context of our example with a 2x2 matrix, after finding the determinant of matrix \(A\), which is 4, and then finding the determinant of its inverse \(A^{-1}\), which is 0.25, we can observe that these values are indeed reciprocals of each other (\(4 \times 0.25 = 1\)). This principle is not only intriguing but also universally applicable for all non-singular square matrices, reinforcing the interconnectedness of the determinant and its inverse.