Problem 24 - Precalculus Mathematics 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 is invertible.

Step 1: Identify the matrix

The matrix given for this problem is a 3x3 matrix:\[\begin{bmatrix}0 & -1 & 0 \2 & 6 & 4 \1 & 0 & 3\end{bmatrix}\]We need to find its determinant to determine if it has an inverse.

Step 2: Recap the formula for a 3x3 determinant

The determinant of a 3x3 matrix \(A\) with elements \(a_{ij}\) is calculated using the formula:\[det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]Substitute the elements of the given matrix into this formula to find the determinant.

Step 3: Substitute values into the determinant formula

Using the elements from the given matrix:- \(a_{11} = 0\), \(a_{12} = -1\), \(a_{13} = 0\)- \(a_{21} = 2\), \(a_{22} = 6\), \(a_{23} = 4\)- \(a_{31} = 1\), \(a_{32} = 0\), \(a_{33} = 3\)The determinant is calculated as:\[0((6)(3) - (4)(0)) - (-1)((2)(3) - (4)(1)) + 0((2)(0) - (6)(1))\]

Step 4: Simplify the determinant expression

First, compute the minor terms in the determinant expression:- First term: \(0((6)(3) - (4)(0)) = 0 \)- Second term: \(-1((2)(3) - (4)(1)) = -1(6 - 4) = -1(2) = -2 \)- Third term: \(0((2)(0) - (6)(1)) = 0\)Thus, the determinant simplifies to: \[0 - (-2) + 0 = 2\].

Step 5: Determine if the matrix is invertible

The determinant of the matrix is \(2\). A matrix is invertible if and only if its determinant is non-zero. Since \(2 eq 0\), the matrix has an inverse.

Key Concepts

Matrix Inverse

Understanding the concept of a matrix inverse is essential in linear algebra. An inverse of a matrix, often denoted as \(A^{-1}\), is a matrix that, when multiplied with the original matrix \(A\), yields the identity matrix. In simpler terms:

• If \(A \times A^{-1} = I\), where \(I\) is the identity matrix, then \(A^{-1}\) is the inverse of \(A\).

Not every matrix has an inverse. One key requirement for a matrix to have an inverse is that its determinant must be non-zero. This means:

• If \(\text{det}(A) eq 0\), the matrix \(A\) is invertible.
• If \(\text{det}(A) = 0\), \(A\) has no inverse, and it is called singular.

In the example given, the determinant was found to be 2, which is non-zero. This confirms that the matrix has an inverse. However, the inverse itself was not calculated as per the task requirements.

3x3 Matrix

A 3x3 matrix is simply a square matrix with three rows and three columns. Each element in the matrix can be identified by its position, represented as \(a_{ij}\), where \(i\) is the row number, and \(j\) is the column number.In the context of the given problem:

• The matrix provided is:\[\begin{bmatrix}0 & -1 & 0 \2 & 6 & 4 \1 & 0 & 3\end{bmatrix}\]

This matrix is 3x3 since it has three rows and three columns. When dealing with such matrices, we often need to compute determinants, find inverses, and solve systems of linear equations. These matrices are prevalent in various applications, including representing transformations in graphics and solving linear algebra problems.

Determinant Formula

The determinant is a special number derived from a square matrix. It offers insights into the properties of the matrix, such as whether it is invertible.To find the determinant of a 3x3 matrix, we use the formula:\[det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]Each term in the formula is associated with the elements in the first row of the matrix and their corresponding minor terms, which are determinants of 2x2 matrices within the original matrix. For example:

• The term \(a_{11}(a_{22}a_{33} - a_{23}a_{32})\) comes from excluding the row and column of \(a_{11}\) and finding the determinant of the resulting 2x2 matrix.

This method ensures that when the determinant is not zero, it implies the matrix has an inverse. Remember, calculating the determinant involves systematic expansion using cofactor expansion, which provides a structured approach to evaluating the matrix determinant efficiently.