Problem 15 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The determinant is 4; thus, the matrix has an inverse.

Step 1: Identify the Matrix

The given matrix is a 3x3 matrix: \[\begin{bmatrix}2 & 1 & 0 \0 & -2 & 4 \0 & 1 & -3\end{bmatrix}\]We need to find the determinant of this matrix to determine if it has an inverse.

Step 2: Use the Determinant Formula for 3x3 Matrices

For a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is given by:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]For our matrix,\(a = 2\), \(b = 1\), \(c = 0\), \(d = 0\), \(e = -2\), \(f = 4\), \(g = 0\), \(h = 1\), \(i = -3\).

Step 3: Calculate Each Component

Substitute the values into each component of the determinant formula:- First term: \( 2((-2)(-3) - 4(1)) = 2(6 - 4) = 2 \times 2 = 4 \).- Second term: \( 1((0)(-3) - 4(0)) = 1(0 - 0) = 0 \).- Third term: \( 0((0)(1) - (-2)(0)) = 0(0 - 0) = 0 \).

Step 4: Find the Total Determinant

Add up all the components:\[4 + 0 + 0 = 4\]The determinant of the matrix is 4.

Step 5: Determine If the Matrix Has an Inverse

Since the determinant is non-zero (4), the matrix has an inverse. A matrix only lacks an inverse if its determinant is zero.

Key Concepts

3x3 matrix

A 3x3 matrix is a system composed of three rows and three columns, forming a grid-like structure with 9 elements. Understanding how to handle such matrices is critical in many areas of mathematics and engineering, especially in linear algebra.
A matrix can contain any number of rows and columns, but a 3x3 matrix is particularly important because it represents transformations in three-dimensional space.

• Each element in the matrix can be identified using two indices: one for the row and one for the column.
• Matrices can be used to solve systems of equations, perform transformations, and determine various properties such as determinants and inverses.

A 3x3 matrix is often denoted as follows:\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \]In our exercise, we're working with:\[\begin{bmatrix} 2 & 1 & 0 \0 & -2 & 4 \0 & 1 & -3 \end{bmatrix} \]Notice how the rows and columns are perfectly aligned, which is crucial when performing operations on the matrix.

matrix inverse

The inverse of a matrix is a key concept, especially in solving linear equations. For a matrix to have an inverse, it must be square, meaning the number of rows is equal to the number of columns, and its determinant must be non-zero.
In our case, the given matrix is a 3x3 matrix, meaning it's square, meeting the first requirement for an inverse. The determinant tells us if the inverse exists and is crucial in calculating it.

• If the determinant is zero, the matrix has no inverse; it's called a singular matrix.
• If the determinant is non-zero, as in our case where it is 4, the matrix is invertible.

Though we're verifying the existence of the inverse and not calculating it, knowing these basics prepares us for future problems where we may need to find the inverse itself.
In practical terms, having a matrix inverse is like having a reciprocating function: it can "undo" what the original matrix does when applied to data or a transformation, which is particularly useful in fields like physics and computer science.

determinant formula

The determinant formula is a specific calculation applied to square matrices that reveals a lot about the matrix's properties. For a 3x3 matrix, the determinant provides essential information about the matrix's invertibility and linear independence of its rows or columns.
Let's break down the formula:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]Here, each letter corresponds to an element in the matrix positioned in our 3x3 layout, as follows:\[\begin{bmatrix}a & b & c \d & e & f \g & h & i \end{bmatrix} \]This formula involves products of specific elements from the matrix and their algebraic cancellations, ensuring every part of the 3x3 grid affects the result.

• First, compute the minor for each element in the first row.
• Next, multiply each minor determinant by its corresponding element from the first row of the matrix and its cofactor sign.
• Finally, add these values to get the total determinant.

For our matrix, we computed to find the determinant as 4, confirming the matrix has an inverse. The determinant directly influences whether a matrix can be used for further applications, such as matrix inversion or solving systems of equations, emphasizing why knowing how to calculate it is crucial.