Question

Find the inverse of the given matrix \(\mathbf{A}\) in each of Exercises 13-24. $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & 0 & -1 \\ 4 & 2 & 1 \\ 2 & 1 & 3 \end{array}\right) $$

Short answer

The inverse of the given matrix \( \mathbf{A} \) is: $$ \mathbf{A^{-1}}=\left(\begin{array}{rrr} \frac{5}{9} & -\frac{5}{9} & -\frac{2}{9} \\ \frac{4}{9} & -\frac{1}{3} & \frac{1}{3} \\ -\frac{2}{9} & \frac{2}{3} & \frac{2}{9} \end{array}\right) $$

Step-by-step solution

Calculate the determinant of the matrix

First, we need to calculate the determinant of the matrix \(\mathbf{A}\). For a 3x3 matrix, the determinant can be calculated as follows: $$ \det(\mathbf{A}) = a(ei - fh) - b(di - fg) + c(dh - eg) $$ where \(\mathbf{A}=\left(\begin{array}{rrr}a & b & c \\\d & e & f \\\g & h & i\end{array}\right)\). In our case, \(\mathbf{A}=\left(\begin{array}{rrr}1 & 0 & -1 \\\4 & 2 & 1 \\\2 & 1 & 3\end{array}\right)\). Therefore, we have a = 1, b = 0, c = -1, d = 4, e = 2, f = 1, g = 2, h = 1, and i = 3. Applying these values to the formula: $$ \det(\mathbf{A}) = 1(6 - 1) - 0 + (-1)(4 - 8) = 5 + 4 = 9 $$ Since the determinant is non-zero, the matrix \(\mathbf{A}\) is invertible.

Find the adjugate of the matrix

The adjugate, or adjoint, of a 3x3 matrix can be found by calculating the cofactors of each element and organizing them into the cofactor matrix, followed by taking the transpose of this matrix. The cofactor is the signed minor of an element of the matrix. Let's find the cofactors of each element: Cofactor matrix: $$ \mathbf{C}=\left(\begin{array}{rrr} (6 - 1) & -(4 - 8) & (2 - 4) \\ -(8 - 3) & (3 - 6) & -(2 - 8) \\ (2 - 4) & -(1 - 4) & (2 - 0) \end{array}\right)=\left(\begin{array}{rrr} 5 & 4 & -2 \\ -5 & -3 & 6 \\ -2 & 3 & 2 \end{array}\right) $$ Transpose of the cofactor matrix: $$ \mathbf{C^T}=\left(\begin{array}{rrr} 5 & -5 & -2 \\ 4 & -3 & 3 \\ -2 & 6 & 2 \end{array}\right) $$ Now, \(\mathbf{C^T}\) is the adjugate (adjoint) of the given matrix \(\mathbf{A}\).

Calculate the inverse of the matrix

The inverse of the matrix \(\mathbf{A}\) can be found by dividing each element of the adjugate by the determinant from Step 1: $$ \mathbf{A^{-1}}=\frac{1}{9}\left(\begin{array}{rrr} 5 & -5& -2 \\ 4 & -3 & 3 \\ -2 & 6 &2 \end{array}\right)=\left(\begin{array}{rrr} \frac{5}{9} & -\frac{5}{9} & -\frac{2}{9} \\ \frac{4}{9} & -\frac{1}{3} & \frac{1}{3} \\ -\frac{2}{9} & \frac{2}{3} &\frac{2}{9} \end{array}\right) $$ Thus, the inverse of the given matrix \(\mathbf{A}\) is: $$ \mathbf{A^{-1}}=\left(\begin{array}{rrr} \frac{5}{9} & -\frac{5}{9} & -\frac{2}{9} \\ \frac{4}{9} & -\frac{1}{3} & \frac{1}{3} \\ -\frac{2}{9} & \frac{2}{3} & \frac{2}{9} \end{array}\right) $$

Key concepts

Determinant of a Matrix

Understanding the determinant of a matrix is essential for various matrix operations, including finding its inverse. In particular, the determinant is a scalar value that can tell us if a matrix is invertible or not—an essential step when dealing with linear systems. For a 3x3 matrix, the determinant is calculated using a specific formula that involves multiplying and subtracting the products of the elements according to their diagonal relationships. The pattern followed is known as a cross multiplication method leading to the standard formula:

det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

This calculation involves taking each element from the first row and multiplying it by the determinant of the 2x2 matrix that remains after removing the row and column of the element in question. Each of these products is then adjusted by a sign corresponding to its position, leading to some products being subtracted and some added. For a matrix to have an inverse, its determinant must be non-zero; otherwise, the matrix is singular, meaning it does not have an inverse.

Adjugate Matrix

An adjugate matrix, sometimes known as the adjoint matrix, plays a pivotal role in calculating the inverse of a square matrix. To find the adjugate, we first need to calculate the cofactors for each element in the original matrix and organize them into the cofactor matrix. The cofactor of an element is the signed minor, which means it is the determinant of the smaller matrix formed by eliminating the row and column of that element, and then assigned a sign based on its position. To turn the cofactor matrix into the adjugate, simply transpose it. Transposition involves interchanging the rows and columns of the cofactor matrix. The adjugate reflects the unique way in which the elements of a matrix intertwine and work together to establish the matrix's invertibility.

Cofactors of a Matrix

The cofactors of a matrix are determined by the minors, with an added sign. Each minor is the determinant of a smaller matrix derived from the original by removing one row and one column. The cofactor includes a factor of (-1)^(i+j), where i and j stand for the row and column indices of the matrix element in question. This sign pattern creates a checkerboard of plus and minus signs across the matrix.

• The sign ensures that the matrix and its inverse behave correctly in relation to each other.
• Cofactors are instrumental in computing the determinant for larger matrices and finding the adjugate.
• They help in understanding the sensitivity of the determinant to the elements of the matrix.

Cofactors, while being a step in a process for matrix inversion, also provide insight into matrix properties and behaviors.

Inverse of a 3x3 Matrix

Finding the inverse of a 3x3 matrix is a multi-step process. Once we've established that the determinant is not zero, ensuring the matrix is invertible, and computed the adjugate matrix, the inverse can be found by dividing the adjugate matrix by the determinant of the original matrix. Mathematically, this is expressed as:

A^-1 = 1/det(A) * adj(A)

Each element of the adjugate matrix is multiplied by the reciprocal of the determinant, yielding the inverse matrix. This inverse has a powerful property: when it is multiplied by the original matrix, the result is the identity matrix, confirming the accuracy of the inversion process. In practice, matrix inversion allows us to solve linear systems, as the inverse matrix can 'undo' the actions of the original matrix, much like division undoes multiplication in arithmetic.