Problem 36 - Algebra 2

Question

Find the value of each determinant. \(\left|\begin{array}{rr}{3} & {2} \\ {-2} & {4}\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is 16.

Step 1: Identifying the Elements of the Matrix

The determinant we are given is for a 2x2 matrix, which has the form:\[\begin{bmatrix} a & b \c & d \end{bmatrix}\]In this case, our matrix is:\[\begin{bmatrix} 3 & 2 \-2 & 4 \end{bmatrix}\]So, we identify the elements as follows: \(a = 3\), \(b = 2\), \(c = -2\), and \(d = 4\).

Step 2: Applying the Determinant Formula for a 2x2 Matrix

The formula for finding the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \c & d \end{bmatrix}\) is given by:\[ad - bc\]We apply this formula to the elements we identified in Step 1.

Step 3: Calculating the Determinant

Substitute the values of \(a\), \(b\), \(c\), and \(d\) into the determinant formula:\[\text{Determinant} = (3)(4) - (2)(-2)\]Calculate each part separately:\[(3)(4) = 12\]\[(2)(-2) = -4\]Substitute these back into the formula:\[12 - (-4)\]

Step 4: Simplifying the Expression

Simplifying \(12 - (-4)\) means converting the subtraction of a negative to addition:\[12 + 4 = 16\]Thus, the determinant is 16.

Key Concepts

2x2 matrix

In linear algebra, a 2x2 matrix is a square arrangement of numbers with two rows and two columns. The matrix is often represented in brackets like this:

• First row: [a b]
• Second row: [c d]

Here, each number, known as a matrix element, belongs in its unique position within the matrix. The order of the numbers matters because it affects calculations like finding the determinant. In our example, the matrix is represented as \[\begin{bmatrix} 3 & 2 \ -2 & 4 \end{bmatrix}\]Each element has a specific symbol: \(a = 3\), \(b = 2\), \(c = -2\), and \(d = 4\). These elements are key to calculating various properties of the matrix, including the determinant.

determinant formula

The determinant of a 2x2 matrix is a special number calculated from the elements of the matrix. This number can tell you whether a matrix is invertible and, in geometry, can help determine area or volume when matrices are used to describe space transformations.
The formula to find the determinant for a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is written as:

• \(ad - bc\)

This means you multiply the top left and bottom right elements \(a\) and \(d\), subtract the product of the top right and bottom left \(b\) and \(c\). This formula is straightforward for 2x2 matrices, making it easy to compute even manually.

matrix elements

Matrix elements are the individual numbers in the matrix. They define the basic structure and behavior of the matrix. In the formula \(ad - bc\), matrix elements are represented by \(a, b, c,\) and \(d\).

• The element \(a\) is located at the first row, first column.
• The element \(b\) is in the first row, second column.
• The element \(c\) sits in the second row, first column.
• And \(d\) is in the second row, second column.

Knowing the positions and values of these elements is important as they are used in operations like finding the determinant. The determinants rely heavily on these values, particularly their order and placement.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. When you're working with matrices and their determinants, algebra is the tool used to process the matrix elements into meaningful results like the determinant number.
In our example, algebra allows us to compute the determinant by applying the straightforward formula \(ad - bc\). It lets us substitute variables, solve equations, and simplify expressions. This process includes recognizing the pattern:

• First stepping through calculations such as \((3)(4) = 12\)
• And \((2)(-2) = -4\)

Next is simplifying the expression, changing the tricky subtracting a negative into adding numbers. Easy substitutions and simplifications are core techniques in algebra that help solve complex problems systematically.