Problem 4 - Algebra and Trigonometry

Question

\(1-8\) Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{rr}{-2} & {1} \\ {3} & {-2}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is 1.

Step 1: Identify the Matrix

The matrix given in the problem is a 2x2 matrix:\[ A = \begin{bmatrix} -2 & 1 \ 3 & -2 \end{bmatrix} \] where the elements are \(a = -2\), \(b = 1\), \(c = 3\), and \(d = -2\).

Step 2: Use the Formula for 2x2 Matrix Determinant

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \det(A) \) is given by:\[ \det(A) = ad - bc \]

Step 3: Substitute the Values into the Formula

Substitute the values from the matrix into the determinant formula:\[ \det(A) = (-2)(-2) - (1)(3) \]

Step 4: Simplify the Expression

Calculate the products and simplify:\[ \det(A) = 4 - 3 = 1 \]

Step 5: Conclusion

The determinant of the matrix is \(1\). Since the determinant is non-zero, the matrix has an inverse.

Key Concepts

Understanding the 2x2 Matrix

A 2x2 matrix is a mathematical representation that consists of two rows and two columns. It is one of the simplest forms of a matrix, which makes it an excellent starting point for learning about matrices and their properties. In our example, we are dealing with the matrix:
\[\begin{bmatrix} -2 & 1 \ 3 & -2 \end{bmatrix}\]This matrix includes four elements: each entry resides in a specific position defined by its row and column, often denoted as:

• First row, first column: \(a = -2\)
• First row, second column: \(b = 1\)
• Second row, first column: \(c = 3\)
• Second row, second column: \(d = -2\)

Matrices like these are essential in various areas such as solving systems of equations and transformations in geometry.

Exploring the Inverse of a Matrix

The inverse of a matrix, if it exists, is a crucial concept in linear algebra. It can be thought of as the "opposite" of a matrix, much like how the number 3 has an inverse of \(\frac{1}{3}\) in multiplication. For a 2x2 matrix, the inverse is denoted as \(A^{-1}\) and can only be determined if the determinant of the matrix is non-zero, which we have in this case as:
\[\det(A) = 1\]The formula to find the inverse of a 2x2 matrix \(A\) is:
\[A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]For our matrix:

• The element \(d\) becomes \(-2\)
• The element \(b\) becomes \(-1\)
• The element \(c\) becomes \(-3\)
• The element \(a\) becomes \(-2\)

Therefore, \(A^{-1}\) can be calculated using the elements of the original matrix.

Matrix Determinants Formula Expanded

The determinant of a matrix is a special number that can be derived from its elements. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated with a straightforward formula:
\[\det(A) = ad - bc\]The determinant provides whether a matrix is invertible or not. A non-zero determinant means the matrix is invertible, like in our instance where:
\[\det(A) = (-2)(-2) - (1)(3) = 4 - 3 = 1\]This result confirms that our matrix can indeed have an inverse. Recognizing the determinant helps in many applications such as scaling transformations and solving linear systems, where having a non-zero determinant ensures unique solutions exist.