Problem 25 - Algebra 2

Question

Find the value of each determinant. $$ \left|\begin{array}{rrr}{8} & {-9} & {0} \\ {1} & {5} & {4} \\ {6} & {-2} & {3}\end{array}\right| $$

Step-by-Step Solution

Answer

The value of the determinant is \(-5\).

Step 1: Understand Determinant of 3x3 Matrix

To find the determinant of a 3x3 matrix, we use the formula: \[ \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \] where the matrix is: \[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \]

Step 2: Identify Values from the Matrix

Compare the given matrix \[ \begin{bmatrix} 8 & -9 & 0 \ 1 & 5 & 4 \ 6 & -2 & 3 \end{bmatrix} \] with the formula matrix to identify: \( a = 8, b = -9, c = 0, d = 1, e = 5, f = 4, g = 6, h = -2, i = 3 \).

Step 3: Compute Individual Components

Calculate each of the smaller determinants: 1. \( ei - fh = 5 \times 3 - 4 \times (-2) = 15 + 8 = 23 \). 2. \( di - fg = 1 \times 3 - 4 \times 6 = 3 - 24 = -21 \). 3. \( dh - eg = 1 \times (-2) - 5 \times 6 = -2 - 30 = -32 \).

Step 4: Substitute Back into the Determinant Formula

Using the smaller determinant values in the main formula:\[d = 8 \times 23 - (-9) \times (-21) + 0 \times (-32)\]Calculate each part:1. \( 8 \times 23 = 184 \).2. \( -9 \times (-21) = 189 \).3. \( 0 \times (-32) = 0 \).

Step 5: Calculate Final Result

Simplify the expression:\[184 - 189 + 0 = -5\]Thus, the determinant of the matrix is \(-5\).

Key Concepts

3x3 Matrix

A 3x3 matrix is a collection of numbers organized into three rows and three columns. It can be presented in a rectangular array usually enclosed within square brackets. Each number within the matrix is called an element. Here's an example matrix:\[\begin{bmatrix}a & b & c \d & e & f \g & h & i \\end{bmatrix}\]Understanding how values are arranged in a 3x3 matrix is very important for performing operations like calculating the determinant. The position of each element is defined by its row and column within the matrix. Learning the layout of a 3x3 matrix serves as a foundation for further topics like finding determinants and performing matrix multiplication.

Determinant Formula

Finding the determinant of a 3x3 matrix is a crucial operation in linear algebra. The determinant is a single number calculated from a matrix and provides insight into matrix properties, such as invertibility. To find the determinant of a 3x3 matrix, use the formula:\[\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]Breaking it down:

• Identify and note the elements from your 3x3 matrix, labeling them as \(a, b, c, d, e, f, g, h,\) and \(i\).
• Calculate three products, which are smaller determinants: \(ei - fh\), \(di - fg\), and \(dh - eg\).
• Plug each product into the formula, multiplying by the corresponding elements \(a, b,\) and \(c\), and apply subtraction and addition as indicated by the formula.

This step-by-step process is designed to simplify what may seem complex, transforming it into manageable pieces.

Matrix Multiplication

Matrix multiplication involves creating a new matrix by multiplying corresponding elements from two existing matrices. Although different from calculating a determinant, understanding matrix multiplication is essential in understanding matrices as a whole and their applications. Here's a simple way to consider matrix multiplication:

• To multiply two matrices, ensure the number of columns in the first matrix equals the number of rows in the second matrix.
• Each element of the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the column from the second matrix.
• It’s a systematic process and often involves a large volume of calculation, making attention to detail crucial.

Matrix multiplication forms the basis for many advanced topics, like transformations and solving systems of equations using matrices.