Question

Find the inverse of each of the following matrices. (a) \(E=\left[\begin{array}{rrr}4 & -2 & 1 \\ 7 & 3 & 0 \\ 2 & 0 & 1\end{array}\right]\) (c) \(G=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]\) (b) \(F=\left[\begin{array}{rrr}1 & -1 & 2 \\ 1 & 0 & 3 \\ 4 & 0 & 2\end{array}\right]\) \((d) H=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\)

Short answer

(a) Matrix E is invertible with inverse \(E^{-1} = \frac{1}{20} \begin{bmatrix} 3 & 2 & -1 \\ -7 & 4 & 14 \\ -2 & 0 & 8 \end{bmatrix}\); (c) Matrix G is not invertible; (b) Matrix F is invertible with inverse \(F^{-1} = \frac{1}{4} \begin{bmatrix} 0 & 2 & -3 \\ -2 & -2 & 7 \\ 1 & -1 & 1 \end{bmatrix}\); (d) Matrix H is its own inverse: \(H^{-1} = H\).

Step-by-step solution

Is the Matrix Invertible?

To find an inverse, first ensure the matrix is square and has a non-zero determinant. A square matrix is invertible only if its determinant is not zero. Use this test for each matrix.

Inverse of Matrix E

Calculate the determinant of matrix \(E\). If it's non-zero, use the adjugate and determinant formula to find the inverse.Determinant, \( \text{det}(E) = 4(3)(1) + (-2)(0)(2) + (1)(7)(0) - (1)(3)(2) - (-2)(7)(1) - (4)(0)(0) = 12 + 0 + 0 - 6 + 14 + 0 = 20 \).Since \(\text{det}(E) eq 0 \), matrix \(E\) is invertible.The inverse is given by:\[E^{-1} = \frac{1}{20} \begin{bmatrix} 3 & 2 & -1 \ -7 & 4 & 14 \ -2 & 0 & 8 \end{bmatrix}\].

Inverse of Matrix G

Determine if matrix \(G\) has an inverse by calculating its determinant. The determinant is calculated as:Determinant, \( \text{det}(G) = 1(0)(0) + 0(1)(0) + 0(0)(1) - 0 \times 0 \times 1 - 0 \times 1 \times 0 - 0 \times 0 \times 0 = 0 \).Since \( \text{det}(G) = 0 \), matrix \( G \) is not invertible.

Inverse of Matrix F

Calculate the determinant of matrix \(F\). If it's non-zero, proceed to find the inverse using the adjugate and determinant formula.Determinant, \( \text{det}(F) = 1(0)(2) + (-1)(0)(4) + 2(1)(0) - 2(0)(2) - (-1)(1)(4) - (1)(0)(0) = 0 + 0 + 0 - 0 + 4 - 0 = 4 \).Since \(\text{det}(F) eq 0 \), matrix \(F\) is invertible.The inverse is given by:\[F^{-1} = \frac{1}{4} \begin{bmatrix} 0 & 2 & -3 \ -2 & -2 & 7 \ 1 & -1 & 1 \end{bmatrix}\].

Inverse of Matrix H

Matrix \(H\) is known as the identity matrix, \(I\). The identity matrix is always its own inverse regardless of size because multiplying any matrix by the identity matrix returns the original matrix. Therefore, the inverse of matrix \(H\) is itself, i.e., \(H^{-1} = I\):\[H^{-1} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = H\].

Key concepts

Determinant Calculation

To determine if a square matrix is invertible, the first step is calculating its determinant. The determinant provides essential information about the matrix, telling us if the matrix has an inverse or not. The formula for calculating the determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]Calculating determinants can seem challenging at first, but they generally follow this structured approach. For instances of non-zero determinant, as seen with matrices \( E \) and \( F \) in the example, we can proceed to find their inverses. If a determinant is zero, like in matrix \( G \), the matrix is singular and does not have an inverse.

Adjugate Matrix

The adjugate matrix plays a crucial role in finding the inverse of a square matrix. It involves swapping and changing the signs of certain elements of the original matrix. Once you calculate the determinant and ensure it is non-zero, the adjugate is used to compute the inverse.For a matrix \( A \), the adjugate (often denoted as \( \text{adj}(A) \)), can be found by:- Transposing the cofactor matrix of \( A \).- The cofactor of each term \( a_{ij} \) is calculated by removing the i-th row and j-th column, then computing the determinant of the resulting submatrix.Here's the relationship with the inverse:\[ A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \]This formula shows that with a non-zero determinant, we can solve for the inverse using the adjugate matrix, combining it with the determinant factor.

Identity Matrix

The identity matrix is an essential entity in linear algebra. A square identity matrix is a matrix with ones on the diagonal and zeros elsewhere, represented as \( I_n \) for an \( n\times n \) matrix.Here's what makes identity matrices important:- Multiplying any matrix \( A \) by an identity matrix \( I \) results in \( A \). It acts like the number 1 in multiplication.- Identity matrices are their own inverses. Meaning \( I^{-1} = I \).In practical terms, for matrix \( H \), which is an identity matrix, its inverse is itself. Understanding identity matrices assists in grasping how operations involving inverses work, especially with square matrices.

Matrix Invertibility

Matrix invertibility is a fundamental concept in matrices that determines whether a matrix can be "reversed" through multiplication. A matrix is said to be invertible or non-singular if its determinant is non-zero.To check invertibility:- Verify the matrix is square (same number of rows and columns).- Calculate its determinant.- Ensure the determinant is not zero.When a matrix is invertible, you can uniquely find a matrix \( B \) such that \( AB = BA = I \), where \( I \) is the identity matrix.A clear understanding of invertibility helps in solving systems of linear equations, transforming geometric spaces, and numerous applications in engineering and science. Matrix \( E \) and \( F \) from the example are invertible, providing solutions to their respective equations, while matrix \( G \) is not due to a zero determinant.