Problem 24 - Algebra 2

Question

Find the value of each determinant. $$ \left|\begin{array}{rrr}{1} & {5} & {2} \\ {-6} & {-7} & {8} \\ {5} & {9} & {-3}\end{array}\right| $$

Step-by-Step Solution

Answer

The value of the determinant is 21.

Step 1: Understand the Determinant Formula

For a 3x3 matrix with elements \[ \left|\begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array}\right| \]the determinant is calculated using the formula: \[ a(ei - fh) - b(di - fg) + c(dh - eg) \].

Step 2: Identify the Matrix Elements

For the given matrix \[ \left|\begin{array}{ccc} 1 & 5 & 2 \ -6 & -7 & 8 \ 5 & 9 & -3 \end{array}\right| \]assign the elements as follows:- \( a = 1 \), \( b = 5 \), \( c = 2 \)- \( d = -6 \), \( e = -7 \), \( f = 8 \)- \( g = 5 \), \( h = 9 \), \( i = -3 \).

Step 3: Calculate Part of the Determinant

Calculate \( ei - fh \):\[ ei = (-7)(-3) = 21 \]\[ fh = (8)(9) = 72 \]\[ ei - fh = 21 - 72 = -51 \].

Step 4: Calculate Another Part of the Determinant

Calculate \( di - fg \):\[ di = (-6)(-3) = 18 \]\[ fg = (8)(5) = 40 \]\[ di - fg = 18 - 40 = -22 \].

Step 5: Calculate the Final Part of the Determinant

Calculate \( dh - eg \):\[ dh = (-6)(9) = -54 \]\[ eg = (-7)(5) = -35 \]\[ dh - eg = -54 - (-35) = -54 + 35 = -19 \].

Step 6: Compute the Determinant

Substitute the calculated parts back into the determinant formula:\[ a(ei - fh) - b(di - fg) + c(dh - eg) = 1(-51) - 5(-22) + 2(-19) \]Calculate:\[ 1(-51) = -51 \]\[ -5(-22) = 110 \]\[ 2(-19) = -38 \]Add these values:\[ -51 + 110 - 38 = 21 \].

Step 7: Write the Final Answer

The value of the determinant is \( 21 \).

Key Concepts

3x3 matrix

A 3x3 matrix is a square arrangement of numbers made up of 3 rows and 3 columns. Mathematicians use these matrices to solve a wide array of problems, especially within linear algebra. Each number inside the matrix is known as an element. Essentially, a 3x3 matrix can be thought of as a grid containing nine elements in total.

Matrices are important because they can represent systems of equations, transformations, and various other mathematical objects. In many mathematical scenarios, understanding the matrix's structure helps solve related problems, like finding determinants or solving equations.

In the context of this exercise, the 3x3 matrix being analyzed has elements that you can arrange conveniently to visualize patterns and perform calculations. Knowing the formula to compute the determinant effectively helps manage the specific operations needed.

linear algebra

Linear algebra is a branch of mathematics concerning vectors, matrices, and linear transformations. It's essential in various fields such as computer science, engineering, physics, and economics. Within this realm, matrices and their determinants play a vital role.

Of particular interest in linear algebra is the calculation of determinants, especially for square matrices like a 3x3 matrix. The determinant offers insight into various properties of the matrix, such as whether a matrix is invertible and the volume distortion in linear transformations.

To calculate the determinant of a 3x3 matrix, you rely on a specific formula that combines the products of its elements. Understanding how to work with these formulas is key in linear algebra, as it allows students to unravel more complex problems and theories derived from these basic operations.

matrix elements

Each number within a matrix is called an element. In our example of a 3x3 matrix, these elements were labeled as follows:

• Row 1: elements are denoted by \(a = 1\), \(b = 5\), \(c = 2\).
• Row 2: elements include \(d = -6\), \(e = -7\), \(f = 8\).
• Row 3: elements are \(g = 5\), \(h = 9\), \(i = -3\).

The determinant of the matrix depends on these elements and their position in the matrix. The computation involves strategic pairing and cross-multiplication of these matrix elements, followed by addition and subtraction.

This procedure emphasizes the importance of matrix elements, especially in mathematical operations like calculating determinants. Understanding each element's role aids in mastering the mechanics behind matrix calculations, crucial for diving deeper into subjects such as eigenvalues or linear mappings.