Problem 19 - Precalculus: Mathematical for Calculus

Question

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

Identify the given matrix as a 3x3 matrix:\[ A = \begin{bmatrix} 30 & 0 & 20 \ 0 & -10 & -20 \ 40 & 0 & 10 \end{bmatrix} \]

Step 2: Find the Determinant of Matrix

To find the determinant of a 3x3 matrix \(A\), use the formula:\[ \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]For our matrix, assign:\[ a = 30, b = 0, c = 20 \d = 0, e = -10, f = -20 \g = 40, h = 0, i = 10 \]Plug these values into the formula:\[ \text{det}(A) = 30((-10)(10) - (-20)(0)) - 0(0(10) - (-20)(40)) + 20(0(0) - (-10)(40)) \]Simplify inside the brackets:\[ = 30(-100 + 0) + 20(0 + 400) \]\[ = 30(-100) + 20(400) \]\[ = -3000 + 8000 \]\[ = 5000 \]

Step 3: Determine the Invertibility

A matrix has an inverse if its determinant is not zero. Here, the determinant is \(5000\). Since \(5000 eq 0\), the matrix has an inverse.

Key Concepts

Matrix Inverse

Understanding the concept of a matrix inverse is crucial in linear algebra. A matrix is said to have an inverse if there exists another matrix that, when multiplied with the original, results in the identity matrix. For a square matrix like a 3x3 matrix, the identity matrix is a 3x3 matrix with ones on the diagonal and zeros elsewhere.

If you have a matrix \( A \), its inverse is denoted as \( A^{-1} \), and the product \( A \times A^{-1} = I \), where \( I \) is the identity matrix. Not every matrix has an inverse.

For a 3x3 matrix to be invertible, its determinant must be non-zero. Determining whether a matrix has an inverse without actually finding it involves checking this condition. If the determinant is zero, the matrix is singular and does not have an inverse.

Matrix Determinant Formula

The determinant of a matrix provides valuable information about the matrix such as whether it is invertible. For a 3x3 matrix, the determinant is calculated using a specific formula that involves all elements of the matrix.

For a matrix:

• \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \)

The determinant formula is:

• \( \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \)

This formula expands the matrix into smaller determinants, simplifying the calculation through systematic multiplication and subtraction. Calculating determinants is a fundamental skill in matrix algebra, and it's essential for determining the invertibility of the matrix.

3x3 Matrix Algebra

3x3 Matrix algebra involves performing operations on matrices with three rows and three columns. It includes addition, subtraction, multiplication, and finding determinants. Understanding matrix algebra helps solve systems of linear equations, which is a core part of linear algebra.

A 3x3 matrix is represented as:

• \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \)

In algebraic operations, following the order of elements is vital. For multiplication particularly, the rows of the first matrix are multiplied with the columns of the second matrix. Matrix operations follow certain rules, like commutativity in addition, but not in multiplication.

Grasping these concepts helps understand more advanced topics like transformation and eigenvalues.

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is a square matrix that possesses an inverse. Whether a matrix is invertible is determined by its determinant.

An invertible matrix must have a non-zero determinant. In practical terms, this means that the matrix transformation is reversible. Every row and column must be linearly independent, representing unique directions in space, giving the matrix full rank.

Invertibility plays a significant role in various applications such as solving linear systems and computer graphics. Only invertible matrices can be used to perform matrix "division" effectively by multiplying by an inverse.