Problem 20 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The determinant is 60.

Step 1: Understand the Determinant Formula for a 3x3 Matrix

The formula for finding the determinant of a 3x3 matrix is: \[\begin{vmatrix} a & b & c \d & e & f \g & h & i\end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)\]where \(a, b, c, d, e, f, g, h, i\) are the elements of the matrix.

Step 2: Identify the Components of the Matrix

Identify the corresponding elements of the matrix:\(a = -3, \ b = 0, \ c = 6, \d = 6, \ e = 5, \ f = -2, \g = 1, \ h = 4, \ i = 2\)

Step 3: Compute Each Part of the Determinant Formula

First, calculate each of the smaller determinants needed:- \(ei - fh = 5 \times 2 - (-2) \times 4 = 10 + 8 = 18\)- \(di - fg = 6 \times 2 - (-2) \times 1 = 12 + 2 = 14\)- \(dh - eg = 6 \times 4 - 5 \times 1 = 24 - 5 = 19\)

Step 4: Plug Values into the Determinant Formula

Substitute the calculated values into the determinant formula:\[\begin{vmatrix} -3 & 0 & 6 \6 & 5 & -2 \1 & 4 & 2\end{vmatrix} = -3(18) - 0(14) + 6(19)\]

Step 5: Perform the Arithmetic

Calculate the expression:\(-3(18) = -54\)\(0(14) = 0\)\(6(19) = 114\)Combine all parts: \(-54 + 0 + 114 = 60\).

Step 6: State the Final Result

The value of the determinant is 60.

Key Concepts

3x3 matrix

A key concept in linear algebra is the 3x3 matrix. This is simply a square arrangement of numbers with three rows and three columns. Each number in the matrix is known as an 'element'.
Matrix elements are typically arranged in a neat and organized manner, which makes it easier to visually process and manipulate them in equations.
The given matrix in the example is:

• -3 in the first row, first column
• 0 in the first row, second column
• 6 in the first row, third column
• 6 in the second row, first column
• 5 in the second row, second column
• -2 in the second row, third column
• 1 in the third row, first column
• 4 in the third row, second column
• 2 in the third row, third column

When working with a 3x3 matrix, understanding the position of each element allows you to apply various mathematical operations, such as finding the determinant. This is crucial for solving systems of equations and other applications in linear algebra.

determinant formula

The determinant formula for a 3x3 matrix is a mathematical expression that helps in calculating the determinant, a special number associated with the matrix. This number can tell us a lot about the matrix, such as whether it can be inverted or how scaling affects volume in geometric transformations.
The formula for a matrix \(\begin{pmatrix} a & b & c \ n d & e & f \ n g & h & i \\end{pmatrix}\) is:\[a(ei - fh) - b(di - fg) + c(dh - eg)\]Let's break it down:

• \(a(ei - fh)\): Multiply element \(a\) by the difference \(ei - fh\). This is called the minor of \(a\).
• \(-b(di - fg)\): Multiply element \(b\) by the difference \(di - fg\), and subtract. This is the minor of \(b\).
• \(+c(dh - eg)\): Multiply element \(c\) by the difference \(dh - eg\), and add. This is the minor of \(c\).

The calculated value helps in determining the properties of the matrix, such as its invertibility. In our example, this calculation was employed to find that the determinant equals 60.

linear algebra

Linear algebra is a foundational branch of mathematics focused on vectors, vector spaces, and linear transformations. It's crucial in many areas of mathematics, physics, computer science, and engineering.
Central to linear algebra are concepts like matrices and determinants, which are used to solve linear systems of equations, among other tasks. For instance, a system of equations can be succinctly represented using a matrix.
The determinant provides vital information about a matrix:

• If its value is zero, the matrix is singular, meaning there is no unique solution to the associated system of equations.
• If non-zero, the matrix is invertible, indicating a unique set of solutions and confirming certain transformations' properties.

Understanding the processes outlined in the solution, such as computing a determinant, is essential for tackling complex problems in linear algebra and various applications in the sciences.