Problem 23 - Algebra 2

Question

Find the value of each determinant. $$ \left|\begin{array}{lll}{1} & {1} & {1} \\ {3} & {9} & {5} \\ {8} & {7} & {4}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given matrix is -22.

Step 1: Understand the formula for a 3x3 determinant

The determinant of a 3x3 matrix \(A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\) is calculated using the formula: \(\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\).

Step 2: Substitute values into the determinant formula

Using the given matrix \(\begin{bmatrix} 1 & 1 & 1 \ 3 & 9 & 5 \ 8 & 7 & 4 \end{bmatrix}\), substitute the values into the determinant formula: \[\det(A) = 1(9\cdot4 - 5\cdot7) - 1(3\cdot4 - 5\cdot8) + 1(3\cdot7 - 9\cdot8)\].

Step 3: Calculate each minor determinant

Calculate the minor determinants: - \(9\cdot4 - 5\cdot7 = 36 - 35 = 1\),- \(3\cdot4 - 5\cdot8 = 12 - 40 = -28\),- \(3\cdot7 - 9\cdot8 = 21 - 72 = -51\).

Step 4: Substitute minor determinants back into the formula

Substitute the calculated minors back into the original equation from Step 2: \[\det(A) = 1(1) - 1(-28) + 1(-51)\].

Step 5: Simplify the expression to find the determinant

Simplify to calculate the determinant: \[\det(A) = 1 + 28 - 51 = -22\].

Key Concepts

3x3 matrix

A 3x3 matrix is a grid of numbers consisting of three rows and three columns. Each number in the matrix is known as an element. In a matrix, the position of each element is important, typically denoted by \(a_{ij}\), where \(i\) is the row number, and \(j\) is the column number. For instance, in the matrix \[\begin{bmatrix}1 & 1 & 1 \3 & 9 & 5 \8 & 7 & 4 \end{bmatrix}\]we have 1 as the element in the first row and first column, and 9 as the element in the second row and second column. Understanding the arrangement of elements is essential because it influences how operations on the matrix, like finding a determinant, are performed. Using matrices is very common in mathematics, especially in solving systems of equations, transforming geometrical shapes, and representing various mathematical concepts.

determinant formula

The determinant formula for a 3x3 matrix provides us with a way to extract a single value that represents properties of the matrix. A matrix \(A\) with elements \(a, b, c, d, e, f, g, h,\) and \(i\), can represent this using the formula: \[det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This formula might look daunting at first, but it is a systematic approach to breaking down the matrix into smaller, more manageable pieces known as minor determinants (which we’ll get into later).

• The formula takes each element from the first row and multiplies it by the determinant of a smaller 2x2 matrix (created by removing the row and column of the element being multiplied).
• It’s a methodical way that relies on the expansion of the matrix across the first row, similar to peeling layers.

Calculating determinants is crucial in many areas of math and science, including calculus, linear algebra, and physics.

matrix calculation

Matrix calculation involves various operations, such as addition, multiplication, and finding determinants, that allow us to manipulate the grid of numbers within a matrix. When calculating the determinant, we specifically aim for a value that reflects certain characteristics of the matrix.The steps for calculating a 3x3 determinant are straightforward if followed carefully:

• First, apply the determinant formula by selecting an element from the first row.
• Next, compute the minor determinant for each selected element.
• Finally, substitute these minor results into the determinant formula and simplify the expression to get the determinant value.

For instance, in the exercise, the matrix \[\begin{bmatrix}1 & 1 & 1 \3 & 9 & 5 \8 & 7 & 4 \end{bmatrix}\]was substituted into our determinant formula, leading us to the final calculation. Understanding these steps allows us to solve problems efficiently.

minor determinant

A minor determinant is a term used in matrix calculations to refer to the determinant of a smaller matrix that remains after deleting a row and a column from the said larger matrix. This smaller 2x2 determinant is pivotal in breaking down and calculating the determinant of a 3x3 matrix.Let's consider how it's done:

• Choose an element from the first row of the original matrix, say \(a_{11}\).
• Remove the row and column of this element to form a 2x2 matrix.
• Calculate the determinant of this smaller matrix to find the minor determinant.

For example, to find the minor determinant correlated to the first element (1) in our given matrix, remove the first row and first column, resulting in the matrix \(\begin{bmatrix} 9 & 5 \ 7 & 4 \end{bmatrix}\), whose determinant would be \(9 \cdot 4 - 5 \cdot 7 = 1\).These minor determinants are the building blocks in our determinant formula for larger matrices, helping us systematically solve for the ultimate value of the determinant. By understanding minor determinants, students can manage calculations of larger matrices with more confidence.