Problem 23 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{llll}{1} & {2} & {0} & {3} \\ {0} & {1} & {1} & {1} \\\ {0} & {1} & {0} & {1} \\ {1} & {2} & {0} & {2}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrix does not have an inverse because it is singular (determinant is zero).

Step 1: Determine the Matrix Order

The given matrix is 4x4, indicating there are 4 rows and 4 columns. For a matrix to have an inverse, it must be square (it is) and its determinant must be non-zero.

Step 2: Calculate the Determinant

We need to find the determinant of the matrix. If the determinant is zero, the matrix does not have an inverse. Calculation of determinant for a 4x4 matrix involves expansion by minors.

Step 3: Check if Matrix is Full Rank

Instead of calculating the full determinant, notice that two rows are linearly dependent (the fourth row is just twice the first row minus one) which makes the rank less than full (less than 4).

Step 4: Conclude on Invertibility

Since the determinant is zero due to linear dependence of rows, the matrix is not of full rank and therefore, it does not have an inverse.

Key Concepts

Matrix Determinants

Understanding matrix determinants is crucial when dealing with matrix inversion. A determinant is a special number that is derived from a square matrix. It tells us important properties about the matrix. For a matrix to have an inverse, its determinant must be non-zero.
The process to find a determinant varies with the size of the matrix. For a 4x4 matrix, this usually involves the method of expansion by minors, which can be complex. However, there's a smarter way to utilize the properties of determinants. If through any shortcut, such as identifying zero rows, duplicate rows, or obvious linear dependencies, we find the determinant to be zero, the matrix has no inverse.
Determinants can give us information not only about invertibility but also about the geometric properties of a transformation represented by the matrix, like scaling and rotation. In practical calculations, the conclusion of a zero determinant quickly suggests that a matrix is singular, meaning it is not invertible.

Linear Dependence

Linear dependence among rows or columns of a matrix is a critical concept when determining its inverse. Typically, linear dependence means that one row or column is a linear combination of other rows or columns. This relationship affects the matrix rank as well as its determinant.
In the provided matrix exercise, the fourth row is linearly dependent on the first row. Specifically, it can be shown as twice the first row minus one. This linear dependency implies that the matrix columns (or rows) don't span the entire vector space, making the matrix's determinant zero.
Linear dependence greatly simplifies the process of determining whether a matrix can be inverted. When dependencies exist, calculation shortcuts can be taken to deduce properties and characteristics of matrices without the need for full computation of their determinants.

Matrix Rank

Matrix rank is an essential concept that indicates the number of independent rows or columns within a matrix. It provides insight into the solution of linear equations represented by the matrix and is indicative of linear dependence.
A matrix with full rank (equal to the number of rows or columns) ensures that it has no linear dependencies, and often, a non-zero determinant. Such matrices are invertible. In our problem's matrix, due to the observed linear dependence, the matrix does not have full rank. Specifically, the actual rank is less than 4, which matches the presence of linearly dependent rows.
Rank can be determined by row reducing the matrix to its echelon form or simply identifying linear dependencies, as was the case here. Matrices that lack full rank, often called "rank-deficient" matrices, are not invertible, underscoring why understanding matrix rank is crucial in the study of linear algebra.