Problem 14 - Precalculus with Limits

Question

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} 2 & -3 \\ -6 & 9 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the given matrix is 0.

Step 1: Identify matrix elements

Here is the matrix: \( \left[ \begin{array}{r} 2 & -3 \ -6 & 9 \end{array} \right] \) \nSymbolizing the elements of the matrix with letters: \( \left[ \begin{array}{r} a & b \ c & d \end{array} \right] \), we can assign: \( a = 2, b = -3, c = -6, d = 9 \)

Step 2: Apply the determinant formula

Apply the formula \(ad - bc\) using the values of \(a, b, c, d\) identified in the previous step. The computation will look like this: \( (2 * 9) - (-3 * -6) \)

Step 3: Compute the determinant

Perform the operations inside the brackets first. This results in: \( (18) - (18) = 0 \)

Key Concepts

Determinant Formula

Understanding the determinant of a matrix is essential for various applications in mathematics and engineering, such as solving systems of linear equations. For a 2x2 matrix, the determinant can be calculated using the formula:
\[ det(A) = ad - bc\]where the matrix \[A\] is given by \[ \left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\].
In this formula, 'a' and 'd' are the elements of the leading diagonal, and 'b' and 'c' are the elements of the other diagonal. To find the determinant, multiply 'a' by 'd', multiply 'b' by 'c', and subtract the second product from the first.

Example:

If we take the matrix from the exercise,\[ \left[ \begin{array}{cc} 2 & -3 \ -6 & 9 \end{array} \right]\],using the determinant formula, we calculate: \[ (2 \cdot 9) - (-3 \cdot -6) \]. After simplifying, we get \[ 18 - 18 = 0 \], hence the determinant is zero.

Matrix Elements

Matrix elements are the individual values that make up a matrix. These elements are usually denoted by a letter and an accompanying subscript that represents their position in the matrix.
For a 2x2 matrix given as\[ \left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\],' a' is situated in the first row and first column, 'b' is in the first row and second column, 'c' is in the second row and first column, and 'd' is in the second row and second column.To solve for the determinant, it is crucial to correctly identify and assign each element. This step is verified in the exercise solution, where we assign: \( a = 2, b = -3, c = -6, d = 9 \). Knowing the value of each matrix element is the first step to correctly applying the determinant formula.

Matrix Operations

Matrix operations include a variety of procedures one can perform with matrices, such as addition, subtraction, multiplication, and finding the determinant. These operations follow specific rules that differ from standard arithmetic. For example, matrix multiplication is not commutative, meaning that \(A \cdot B eq B \cdot A\) in most cases.When finding the determinant of a 2x2 matrix, the operation involves multiplication and subtraction of the elements, as shown in the determinant formula. After identifying the matrix elements, you conduct the operations as indicated by the formula.

Attention to Signs:

It is crucial to pay attention to the signs of the matrix elements. As seen in the step-by-step solution for the exercise, the negative signs in the matrix elements 'b' and 'c' lead to a determinant of zero when the formula is applied correctly: \( (2 \cdot 9) - (-3 \cdot -6) \).Carefully conducting these operations is fundamental to accurately compute the determinant and to further understand more complex aspects of matrices.