Problem 16 - Precalculus: Mathematical for Calculus

Question

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{rrr} 0 & -1 & 0 \\ 2 & 6 & 4 \\ 1 & 0 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 2; the matrix has an inverse.

Step 1: Describe the Matrix

We are given a 3x3 matrix to analyze. The matrix is as follows:\[\begin{bmatrix}0 & -1 & 0 \2 & 6 & 4 \1 & 0 & 3 \end{bmatrix}\]

Step 2: Find the Determinant

To determine whether the matrix has an inverse, we must find its determinant. The formula for the determinant of a 3x3 matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is:\[a(ei - fh) - b(di - fg) + c(dh - eg)\]Substitute the matrix values:\[det = 0(6 \times 3 - 4 \times 0) - (-1)(2 \times 3 - 4 \times 1) + 0(2 \times 0 - 6 \times 1)\]Simplify the expression:\[det = 0 - (-1)(6 - 4) + 0\]\[det = 0 - (-1)(2) = 2\]

Step 3: Determine if an Inverse Exists

A matrix has an inverse if its determinant is not zero. In this case, the determinant is \(2\). Since \(2 eq 0\), the matrix has an inverse.

Key Concepts

Inverse Matrix

An inverse matrix is a concept in matrix algebra that essentially acts as the "undo" operation for matrices, much like division is to multiplication in arithmetic. If you multiply a matrix by its inverse, you end up with the identity matrix, which is the equivalent of "one" in matrix operations. However, not all matrices have inverses. A matrix must be a square matrix, meaning it has the same number of rows and columns, and its determinant must not be zero.

• The process to determine the inverse involves using the formula that requires calculating the determinant and the adjugate of the matrix.
• If a matrix's determinant is zero, the matrix is said to be singular and it does not have an inverse.

In the given exercise, after finding the determinant of the matrix to be 2, which is not zero, we conclude that the matrix does indeed have an inverse. Always remember, the determinant is the key indicator of whether an inverse exists or not.

3x3 Matrix

A 3x3 matrix is a specific type of matrix that contains three rows and three columns. This is a square matrix because the number of rows is equal to the number of columns. Square matrices, like 3x3 matrices, are crucial in linear algebra because they can have inverses, and their determinants can be computed.

• The layout of elements in a 3x3 matrix makes it possible to apply various matrix operations such as addition, multiplication, finding determinants, and calculating inverses.
• Notably, 3x3 matrices appear often in problems related to systems of linear equations and transformations.

When working with 3x3 matrices, practice is essential to become comfortable with the calculations involved, especially those that determine if the matrix has an inverse. The exercise encourages you to apply the determinant formula, reinforcing the fundamental operations necessary for understanding 3x3 matrices thoroughly.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and the operations that can be applied to them. It is particularly useful for solving systems of linear equations and is widely used in various scientific disciplines, including physics, computer science, and engineering.

• Basic operations in matrix algebra include addition, subtraction, multiplication, and finding the determinant and inverse of matrices.
• Understanding these operations is essential for manipulating matrices efficiently and accurately to solve larger, more complex problems.

In the given problem, matrix algebra is applied by calculating the determinant of a 3x3 matrix to verify if it has an inverse. The determinant calculation shows that the matrix is invertible. Being proficient in matrix algebra opens doors to understanding more complex mathematical concepts and solving practical problems in multiple areas.