Problem 9 - Algebra 2

Question

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rr}{-2} & {0} \\ {2} & {-1}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 2.

Step 1: Recall the formula for the determinant of a 2x2 matrix

For a 2x2 matrix \[\begin{array}{cc}a & b \ c & d\end{array}\], the determinant is calculated as \(ad - bc\).

Step 2: Identify the elements of the matrix

For the given matrix \[\begin{array}{rr}{-2} & {0} \ {2} & {-1}\end{array}\], the elements are \(a = -2\), \(b = 0\), \(c = 2\), and \(d = -1\).

Step 3: Calculate the determinant

According to the formula, the determinant of the given matrix is \(ad - bc = (-2)\cdot(-1) - (0)\cdot(2) = 2 - 0 = 2\).

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple square grid of numbers consisting of 2 rows and 2 columns. It is the most basic form of a matrix that still allows for interesting properties and operations, such as calculating determinants, to be observed. In our example, the 2x2 matrix is expressed as:

\[\begin{array}{cc}-2 & 0 \2 & -1\end{array}\]This notation is a compact way to present all the information we need to perform further operations on the matrix.

Matrix Elements

The matrix elements are the individual values or numbers that make up the matrix. Each number is referred to as an element or entry of the matrix. For example, in a 2x2 matrix, there are four elements, usually named using letters such as 'a', 'b', 'c', and 'd'. These elements are positioned in the matrix as:

\[\begin{array}{cc}a & b \c & d\end{array}\]In our specific exercise, these elements are identified as \( a = -2 \), \( b = 0 \), \( c = 2 \), and \( d = -1 \).

Matrix Operations

Matrix operations encompass a variety of arithmetic procedures applied to matrices, including addition, subtraction, multiplication, and finding the determinant among others. The determinant is a special value that can be calculated from a square matrix. It provides important information about the matrix, such as whether it is invertible or what is its scale factor in transformations. To find the determinant of a 2x2 matrix, we use the simple formula \( ad - bc \).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together represent a particular number or value. In the context of matrix operations, the determinant formula \( ad - bc \) is an example of an algebraic expression. It involves multiplication and subtraction of the matrix elements. When you substitute the elements of our given 2x2 matrix into this expression, you receive the solution to our problem which, after applying the arithmetic, gives us a determinant value of 2.