Problem 10 - Precalculus: Functions and Graphs

Question

Find the determinant of the matrix. $$\left[\begin{array}{rr} 6 & 4 \\ -3 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 24.

Step 1: Identify the Elements of the Matrix

The given matrix is \[\begin{bmatrix} 6 & 4 \ -3 & 2 \end{bmatrix}\].The elements of the matrix are: \( a = 6 \), \( b = 4 \), \( c = -3 \), and \( d = 2 \).

Step 2: Recall the Formula for the Determinant of a 2x2 Matrix

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

Step 3: Substitute the Values into the Determinant Formula

Using the formula \( ext{det} = ad - bc \), substitute the known values:\( a = 6 \), \( b = 4 \), \( c = -3 \), \( d = 2 \).Calculate as follows:\[ ext{det} = (6 \times 2) - (4 \times -3)\].

Step 4: Perform the Calculations

Calculate each part:1. \( 6 \times 2 = 12 \).2. \( 4 \times -3 = -12 \).Substitute these results back:\[ ext{det} = 12 - (-12)\]

Step 5: Finalize the Determinant Calculation

Simplify the expression:\( 12 - (-12) = 12 + 12 = 24 \).Thus, the determinant of the matrix is 24.

Key Concepts

2x2 matrix

A 2x2 matrix is one of the simplest forms of matrices in linear algebra, consisting of two rows and two columns. It is often used to represent linear transformations in two-dimensional space. Thinking of a box with four slots can help visualize it: imagine two numbers in the top row and two numbers in the bottom row. A general 2x2 matrix looks like this:\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]where each 'a', 'b', 'c', and 'd' are numerical values that fill the slots of the matrix. These individual values are referred to as the elements of the matrix. Understanding these elements and their arrangement is essential, as they form the basis for further operations such as finding determinants and performing matrix multiplication.

Determinant formula

The determinant of a matrix is a special number that can give us a lot of information about the matrix itself, such as whether it is invertible (non-singular) or not. For a 2x2 matrix, the determinant helps determine area scaling during transformations and can even tell us if a system of equations has a unique solution. The formula for finding the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is:\[\text{det} = ad - bc\]This means you multiply the 'a' element by the 'd' element and then subtract the product of the 'b' element and the 'c' element. The result is the determinant. Knowing this formula is crucial for solving problems involving 2x2 matrices, as it quickly provides the determinant's value, letting you explore further properties of the matrix.

Matrix elements identification

Identifying matrix elements correctly is crucial when you're working with matrices. Each element in a 2x2 matrix has a specific position and notation, commonly labeled with letters such as 'a', 'b', 'c', and 'd'. Each letter corresponds to a particular spot:

• 'a' is the element at the first row and first column.
• 'b' is the element at the first row and second column.
• 'c' is the element at the second row and first column.
• 'd' is the element at the second row and second column.

For example, in the matrix \( \begin{bmatrix} 6 & 4 \ -3 & 2 \end{bmatrix} \), the elements would be identified as:

• \( a = 6 \)
• \( b = 4 \)
• \( c = -3 \)
• \( d = 2 \)

Proper identification of these elements is essential to apply the determinant formula and other matrix operations correctly. Mistaking the position of even one element can lead to entirely different outcomes in your calculations.