Problem 19 - Algebra and Trigonometry

Question

\(15-22\) . Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$ \left[\begin{array}{rrr}{30} & {0} & {20} \\ {0} & {-10} & {-20} \\ {40} & {0} & {10}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant is 5000, and the matrix has an inverse.

Step 1: Identify the matrix elements

The matrix given is \[ A = \begin{pmatrix} 30 & 0 & 20 \ 0 & -10 & -20 \ 40 & 0 & 10 \end{pmatrix} \]. Identify the elements of the matrix as follows: a = 30, b = 0, c = 20, d = 0, e = -10, f = -20, g = 40, h = 0, i = 10.

Step 2: Use the determinant formula for a 3x3 matrix

To find the determinant of a 3x3 matrix, use the formula: \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \].

Step 3: Substitute the elements into the determinant formula

Substitute the elements of the matrix into the determinant formula: \[ \det(A) = 30((-10)(10) - (-20)(0)) - 0(0 \cdot 10 - 40 \cdot -20) + 20(0 \cdot 0 - (-10) \cdot 40) \].

Step 4: Calculate each term separately

Compute each term: \[ei = (-10)(10) = -100 \] \[fh = (-20)(0) = 0 \] \[di = (0)(10) = 0 \] \[fg = 40(-20) = -800 \] \[dh = (0)(0) = 0 \] \[eg = (-10)(40) = -400 \].

Step 5: Simplify and compute the entire determinant expression

Calculate each part of the determinant expression: \[ei - fh = -100 - 0 = -100 \] \[di - fg = 0 - (-800) = 800 \] \[dh - eg = 0 - (-400) = 400 \] Then substitute these back into the determinant expression: \[30 \times (-100) - 0 \times 800 + 20 \times 400 = -3000 + 0 + 8000 = 5000\].

Step 6: Conclude on the matrix inverse existence

The determinant of the matrix is 5000, which is non-zero. Therefore, the matrix has an inverse.

Key Concepts

3x3 matrix

A 3x3 matrix is a grid structure with three rows and three columns, typically represented by a variable like 'A'. Each position within this matrix is occupied by a numerical element. In our example matrix \( A \), each element is a specific number denoting a position in rows and columns, such as \( a = 30 \), \( b = 0 \), and so on.

Understanding the basics of a 3x3 matrix helps in performing operations like addition, subtraction, and multiplication with other matrices.

• Each row contains three elements and each column also contains three elements, leading to a total of nine elements.
• The elements are usually represented in a two-dimensional format inside brackets or parentheses.
• In linear algebra, 3x3 matrices are crucial as they can represent transformations in a three-dimensional space.

When dealing with matrices, it's important to comprehend their structure and how they interact with other algebraic and arithmetic operations.

inverse of a matrix

Finding the inverse of a matrix involves determining another matrix that when multiplied with the original gives an identity matrix. An identity matrix in the 3x3 form looks like \( I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} \). However, a matrix must meet certain criteria to have an inverse.

The most critical criterion is that the matrix must have a non-zero determinant. If the determinant is zero, the matrix is said to be singular and does not have an inverse.

• The formula for the inverse of a 3x3 matrix involves the use of the determinant and the adjugate of the matrix.
• In practical terms, finding the inverse is often required for solving systems of linear equations, among other applications.
• An inverse can help in various transformations and calculations where the undo operation or reverse transformation needs realization.

Calculating the inverse directly can be complex, but understanding its existence is a pivotal step aided by determining if the matrix is invertible.

determinant calculation

The determinant of a matrix is a special number that can be calculated from its elements. For a 3x3 matrix, the formula to determine its determinant \( \det(A) \) is defined as:

\[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Where the letters \( a, b, c, d, e, f, g, h, i \) represent the elements from corresponding positions in the 3x3 matrix.

• The determinant helps characterize the properties of a matrix, like whether it has an inverse.
• A non-zero determinant signifies the matrix is invertible, while a zero determinant implies the matrix does not have an inverse.
• Determinants are also key in applications such as finding areas, volumes, and used in the calculation of eigenvalues.

The computation involves simplifying expressions and involves careful calculations of multiple terms, which then get assembled into the final determinant value, as performed in the original exercise resulting in \( \det(A) = 5000 \).