Problem 57 - Algebra 2

Question

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rrr}{2} & {-1} & {-6} \\ {5} & {0} & {3} \\ {-3} & {2} & {11}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is -8.

Step 1: Identify the matrix

We need to evaluate the determinant of the given 3x3 matrix \( A = \left[\begin{array}{rrr}{2} & {-1} & {-6} \ {5} & {0} & {3} \ {-3} & {2} &{11}\end{array}\right] \).

Step 2: Use the determinant formula for 3x3 matrix

The determinant of a 3x3 matrix \( A = \left[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right] \) is computed as \( \det(A) = a(ei−fh) − b(di−fg) + c(dh−eg) \).

Step 3: Identify matrix components

Identify the components from the matrix: \( a = 2, b = -1, c = -6, d = 5, e = 0, f = 3, g = -3, h = 2, i = 11 \).

Step 4: Calculate individual products

Using the determinant formula, compute the cross-products: - \( ei - fh = 0 \cdot 11 - 3 \cdot 2 = 0 - 6 = -6 \).- \( di - fg = 5 \cdot 11 - 3 \cdot (-3) = 55 + 9 = 64 \).- \( dh - eg = 5 \cdot 2 - 0 \cdot (-3) = 10 - 0 = 10 \).

Step 5: Evaluate the determinant

Substitute the products back into the determinant formula to calculate: \( \det(A) = 2(-6) - (-1)(64) - 6(10) \).This simplifies to \( \det(A) = -12 + 64 - 60 \).

Step 6: Final calculation

Simplify the expression from Step 5: \( \det(A) = -12 + 64 - 60 = -8 \).

Key Concepts

Matrix Algebra

Matrix algebra is a powerful mathematical tool that allows us to perform various operations on matrices. These operations include addition, subtraction, multiplication, and finding determinants, among others.

Matrix algebra is unique as these operations follow specific rules that differ from traditional arithmetic. For instance, the multiplication of matrices is not commutative, meaning that the order in which you multiply matrices matters. This is different from regular numbers where, for example, 2 times 3 is the same as 3 times 2.

Another key operation in matrix algebra is finding the determinant of a matrix, which is a special number calculated from the elements of a square matrix. This number provides important information about the matrix, such as whether it is invertible (meaning whether or not you can find its inverse). The determinant plays a crucial role in solving systems of linear equations, among other applications.

3x3 Matrix

A 3x3 matrix is a square matrix consisting of 3 rows and 3 columns, making a total of 9 elements.

These elements are arranged in an orderly fashion within the matrix, like this:

• First row with elements: a, b, c
• Second row with elements: d, e, f
• Third row with elements: g, h, i

Knowing the elements’ positions is crucial when performing operations such as matrix multiplication or finding the determinant.

Working with 3x3 matrices can be more complex compared to smaller matrices, but they are widely used in fields such as physics, engineering, and computer science. In these disciplines, 3x3 matrices can represent and solve problems related to spatial relations, structural analysis, and even in graphics rendering. They serve as a stepping stone to understanding more complex matrices and operations.

Determinant Formula

Calculating the determinant of a 3x3 matrix involves a specific formula that utilizes all 9 elements of the matrix. It might seem a bit daunting at first, but understanding the steps can simplify the process.

The formula to compute the determinant of a 3x3 matrix, often written as \( ext{det}(A) \), is as follows:

• \( a(ei−fh) \)
• \(- b(di−fg) \)
• \(+ c(dh−eg) \)

Here, \( a, b, c, d, e, f, g, h, \) and \( i \) represent the respective elements of the matrix.

To make things more straightforward:

• Calculate each of the three products: \( ei−fh \), \( di−fg \), and \( dh−eg \).
• Multiply each product by its corresponding matrix element: \( a, -b, \) and \( c \).
• Sum the results of these three multiplications to get the determinant.

This formula filters through a larger set of interactions within the matrix, distilling them into a single number that characterizes the matrix's properties.