Problem 25 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The determinant is 22, so the matrix is invertible.

Step 1: Recall the Matrix Determinant Formula

For a 4x4 matrix, the determinant can be found using the rule of Sarrus or cofactor expansion. Generally, it's more efficient to use shortcut methods like row reduction but for the purpose of understanding, cofactors will be used here.

Step 2: Select a Row/Column for Expansion

Choose a row or column with the most zeros to simplify. Here, we'll use the first column: \( [1, 0, -1, 1] \).

Step 3: Compute Cofactors for First Column

For each element in the matrix, calculate the minor after removing the element's row and column, and then find the cofactor. The cofactors for the first column are chosen (alternating signs), specifically for 1 (row 1), 0 (row 2), -1 (row 3), and 1 (row 4).

Step 4: Compute the Determinant Using Cofactors

Calculate the determinant as: \[1 \cdot \begin{vmatrix}2 & 0 & 1 \0 & 0 & 2 \6 & 4 & 1\end{vmatrix} - 0 \cdot ... + (-1) \cdot \begin{vmatrix}3 & 3 & 0 \2 & 0 & 1 \6 & 4 & 1\end{vmatrix} + 1 \cdot \begin{vmatrix}3 & 3 & 0 \2 & 0 & 1 \0 & 2 & 1\end{vmatrix}\]Only calculate the non-zero terms.

Step 5: Calculate Each 3x3 Minor

Evaluate each 3x3 determinant (the minors). 1. For the first minor: \[\begin{vmatrix}2 & 0 & 1 \0 & 0 & 2 \6 & 4 & 1\end{vmatrix}\] = 4 \times 1 - 2 \times (0 \cdot 4 - 2 \cdot 6) + 0 = 16.2. For the second non-zero minor: \[\begin{vmatrix}3 & 3 & 0 \2 & 0 & 1 \0 & 2 & 1\end{vmatrix}\] = 3 \times (0 - 2 \cdot 1) = -6.

Step 6: Combine Results to Find Determinant

The full determinant is calculated as: \[1 \cdot 16 + 0 - (-1) \cdot \left(-6\right) = 16 + 6 = 22\]

Step 7: Determine if the Matrix is Invertible

A matrix is invertible if its determinant is non-zero. Here, \(det = 22\), so the matrix is invertible.

Key Concepts

Cofactor Expansion

Cofactor expansion is a powerful method used for finding the determinant of a matrix. It involves choosing a specific row or column to "expand" along. The idea here is to simplify the problem by calculating smaller determinants, known as minors, and then assembling these into the final determinant of the larger matrix.

Each element in the selected row or column is considered individually. To form a minor, remove the row and column that intersect at that element. This leaves you with a smaller matrix. Once you have this minor, calculate its determinant. This value can then be turned into a cofactor by applying alternating signs, starting with a positive sign at the top-left of the matrix. For example, in a 4x4 matrix, if you choose the first column for cofactor expansion, you will consider each element in this column and its corresponding minor.

By calculating and summing the products of all the pairs of elements and their cofactors within the selected row or column, you determine the determinant of the entire matrix. This technique is especially useful for educational purposes as it lays bare the structure and relations within the matrix, even though there may be more efficient computational methods in practice.

Matrix Inversion

Matrix inversion is a process that reveals the 'reversal' of a matrix, much like how division is the reverse of multiplication for numbers. If you have an invertible matrix \( A \), its inverse is denoted as \( A^{-1} \), and it satisfies the equation \( A \cdot A^{-1} = I \), where \( I \) is the identity matrix. Finding the inverse is crucial for solving systems of linear equations and other mathematical computations related to matrices.

Not every matrix can be inverted. The ability to invert a matrix depends on the determinant being non-zero. Without a non-zero determinant, you can't find an inverse matrix because the matrix is said to be singular or non-invertible. Calculating the determinant using methods like cofactor expansion helps indicate whether a matrix can be inverted in the first place.

Matrix inversion, while mathematically fascinating, often requires careful calculation, especially in higher-dimensional matrices. Generally, finding an inverse involves either the cofactor approach or row operations that transform the original matrix into the identity matrix. The latter method, known as Gaussian elimination, tends to be more practical for computational purposes.

Invertible Matrices

Invertible matrices, also known as non-singular matrices, are matrices that possess an inverse. The condition required for a matrix to be invertible is specific: its determinant must not be equal to zero. This criterion serves as a test of a matrix's invertibility before attempting to compute an inverse.

Why is invertibility important? Because it allows us to solve matrix equations of the form \( A \cdot X = B \). For an invertible matrix \( A \), we can multiply both sides of the equation by \( A^{-1} \) to isolate \( X \), giving \( X = A^{-1} \cdot B \). This process is fundamental in many areas of applied mathematics and engineering.

When dealing with larger matrices, invertibility indicates how systems behave. In systems of linear equations, for instance, an invertible matrix ensures that a unique solution exists. Non-invertible matrices typically indicate issues like redundancy among the rows/columns or dependency between them, which in physical terms might imply there is no unique solution or that a force balance is inherently unresolved.