Problem 16 - 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. $$ \left[\begin{array}{rrr}{0} & {-1} & {0} \\ {2} & {6} & {4} \\ {1} & {0} & {3}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant is 2, so the matrix has an inverse.

Step 1: Identify the Matrix

Given the matrix is \[ \begin{bmatrix} 0 & -1 & 0 \ 2 & 6 & 4 \ 1 & 0 & 3 \end{bmatrix} \]. This is a 3x3 matrix.

Step 2: Recall the Determinant Formula for 3x3 Matrices

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

Step 3: Assign Matrix Elements to Variables

Assign the elements of the matrix to each variable in the determinant formula: - \( a = 0 \), \( b = -1 \), \( c = 0 \)- \( d = 2 \), \( e = 6 \), \( f = 4 \)- \( g = 1 \), \( h = 0 \), \( i = 3 \)

Step 4: Substitute and Calculate Each Term

Substitute the values into the determinant formula:1. \( a(ei - fh) = 0(6 \cdot 3 - 4 \cdot 0) = 0 \)2. \( -b(di - fg) = -(-1)(2 \cdot 3 - 4 \cdot 1) = (1)(6 - 4) = 2 \)3. \( c(dh - eg) = 0(2 \cdot 0 - 6 \cdot 1) = 0 \)

Step 5: Add Up All Calculated Terms

The determinant is the sum of terms calculated in Step 4:\[ \text{det}(A) = 0 + 2 + 0 = 2 \]

Step 6: Check for the Inverse Existence

A matrix has an inverse if and only if its determinant is not zero. Since \( \text{det}(A) = 2 \), the matrix is invertible.

Key Concepts

3x3 matrix

A 3x3 matrix is a square matrix made up of three rows and three columns. This is a fundamental type of matrix often encountered in linear algebra and several applied fields like physics, computer graphics, and engineering. Matrices are used to represent linear transformations, and a 3x3 matrix can describe transformations in a three-dimensional space. Each element in the 3x3 matrix is identified by its position within the grid, such as element \(a_{ij}\) where \(i\) represents the row number and \(j\) the column number. Understanding the structure of a matrix is important because it forms the basis for various operations, including addition, multiplication, and finding the determinant. The given matrix in the original exercise is:\[\begin{bmatrix} 0 & -1 & 0 \ 2 & 6 & 4 \1 & 0 & 3 \end{bmatrix}\]Each number within this matrix plays a role in matrix computations. For instance, the element at the first row and second column is \(-1\). Such recognition is significant when performing calculations like finding the determinant or the inverse of the matrix.

inverse matrix

An inverse matrix is essentially the 'reciprocal' of a matrix. For square matrices, just like numbers, finding the inverse is a key operation when solving systems of linear equations. Given a matrix \(A\), its inverse is denoted as \(A^{-1}\). The primary condition for the existence of an inverse matrix is that the original matrix must be square and have a non-zero determinant. If a matrix is invertible, multiplying it with its inverse yields the identity matrix. For example:- If \(A * A^{-1} = I\), then \(A\) is invertible.- If the determinant of \(A\) is zero (det(A) = 0), \(A\) does not have an inverse.In the original exercise, the determinant was calculated as 2, showing that the given matrix has an inverse because the determinant is not zero. Therefore, the matrix can be used for solving equations or systems where finding inverses is necessary, like inverting a transformation or solving linear systems.

matrix determinant calculation

The determinant of a matrix is a scalar value that can provide a wealth of information about the matrix. For a 3x3 matrix, the determinant is computed using a specific formula, and it plays a crucial role in determining whether a matrix is invertible. The formula for a 3x3 matrix \(\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\) is:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This formula may initially seem daunting, but it systematically covers the essential calculations needed to find the determinant:

• First Term: Multiply \(a\) with the difference of products \(ei - fh\).
• Second Term: Subtract \(b\) times the difference \(di - fg\).
• Third Term: Finally, add \(c\) times the difference \(dh - eg\).

In our example, substituting the values from the given matrix into this formula yields a determinant of 2. Since this result is non-zero, we can conclude that the matrix has an inverse, confirming it can be used in computations requiring matrix inversion. Understanding how to calculate determinants is vital in linear algebra, as it allows the exploration of matrix properties and the solution of complex mathematical problems.