Problem 8 - Algebra and Trigonometry

Question

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: Know the formula

To find the determinant of a 2x2 matrix \(\left[\begin{array}{cc}a & b \ c & d\end{array}\right]\), you can use the formula: \[ \text{det}(A) = ad - bc \] where \(a, b, c, \text{ and } d\) are elements of the matrix.

Step 2: Identify the elements

In our given matrix: \(\left[\begin{array}{rr}-2 & 1 \ 3 & -2\end{array}\right]\):- \(a = -2\)- \(b = 1\)- \(c = 3\)- \(d = -2\)

Step 3: Substitute the values into the formula

Using the determinant formula \(ad - bc\):- Compute \(a \, d = (-2) \times (-2) = 4\)- Compute \(b \, c = 1 \times 3 = 3\)

Step 4: Calculate the determinant

Subtract the product \(bc\) from \(ad\):\[\text{det}(A) = 4 - 3 = 1\] The determinant of the matrix is 1.

Key Concepts

2x2 Matrix

A 2x2 matrix is a common type of matrix that is presented in the form of two rows and two columns. Each element in the matrix can appear as a specific number, which together form this compact organization of values.

• The first row contains two numbers which are part of the top half of the matrix.
• The second row completes the bottom half of the matrix with its own two numbers.

For instance, if you look at the example matrix: \[\left[ \begin{array}{rr}-2 & 1 \; 3 & -2 \end{array}\right]\]You can identify four key elements. The elements are often labeled as \(a\), \(b\), \(c\), and \(d\) respectively, ordered from left to right, top to bottom. Understanding the positioning of these values is crucial when you start performing operations like determining the determinant, using every element where it belongs in a formula.

Linear Algebra

Linear Algebra is an area of mathematics that plays a huge role in dealing with linear equations, vectors, and, of course, matrices.

• The major operations and principles of linear algebra become especially useful in the scientific community and across numerous applications, such as computer graphics, machine learning, and more.
• A fundamental aspect of linear algebra involves learning how to manipulate and solve systems with matrices, which includes understanding what the determinant represents.

The determinant, for instance, provides important insights into a matrix. It can give you an idea about some matrix properties like whether the matrix is invertible or not, which is a central concept in linear algebra problems. For a 2x2 matrix, if the determinant is zero, the matrix lacks an inverse and is termed singular. It’s these sorts of calculations that form the backbone of applied and theoretical linear algebra.

Matrix Operations

Matrix operations involve various processes like addition, multiplication, and finding the determinant. Mastery of these operations is key to excelling in mathematics and related fields.Determinants:

• The determinant is a value that can be computed from the elements of a square matrix.
• In a 2x2 matrix, the determinant is found through the formula \(ad - bc\), which involves multiplying and subtracting product terms.

When you work through the determinant of a 2x2 matrix, like our given example, it follows:1. Multiply the diagonal top-left to bottom-right (\(a \, d\)).2. Then multiply the diagonal top-right to bottom-left (\(b \, c\)).3. Lastly, subtract the second product from the first to reach the determinant.This simple process helps to unlock further matrix operations and aids in solving matrix-related problems across various contexts.