Question

Find the inverse of each of the following matrices: \((a) A=\left[\begin{array}{ll}5 & 2 \\ 0 & 1\end{array}\right]\) (b) \(B=\left[\begin{array}{rr}-1 & 0 \\ 9 & 2\end{array}\right]\) \((c) C=\left[\begin{array}{rr}3 & 7 \\ 3 & -1\end{array}\right]\) \((d) D=\left[\begin{array}{ll}7 & 6 \\ 0 & 3\end{array}\right]\)

Short answer

(a) \(A^{-1} = \begin{bmatrix}1/5 & -2/5 \\ 0 & 1\end{bmatrix}\); (b) \(B^{-1} = \begin{bmatrix}-1 & 0 \\ 9/2 & 1/2\end{bmatrix}\); (c) \(C^{-1} = \begin{bmatrix}1/24 & 7/24 \\ 1/8 & -1/8\end{bmatrix}\); (d) \(D^{-1} = \begin{bmatrix}1/7 & -2/7 \\ 0 & 1/3\end{bmatrix}\).

Step-by-step solution

Find the Inverse of Matrix A

Matrix A is \(A=\begin{bmatrix}5 & 2 \ 0 & 1\end{bmatrix}\). For a 2x2 matrix \(\begin{bmatrix}a & b \ c & d\end{bmatrix}\), the inverse exists if the determinant \(ad - bc eq 0\), and the formula is \(A^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b \ -c & a\end{bmatrix}\). Calculate the determinant: \[5 \cdot 1 - 0 \cdot 2 = 5\]Since the determinant is 5, matrix A is invertible. Use the formula:\[A^{-1} = \frac{1}{5}\begin{bmatrix}1 & -2 \ 0 & 5\end{bmatrix} = \begin{bmatrix}1/5 & -2/5 \ 0 & 1\end{bmatrix}\]

Find the Inverse of Matrix B

Matrix B is \(B=\begin{bmatrix}-1 & 0 \ 9 & 2\end{bmatrix}\). Calculate its determinant:\[(-1) \cdot 2 - 0 \cdot 9 = -2\]The determinant is -2, so matrix B is invertible. The inverse is:\[B^{-1} = \frac{1}{-2}\begin{bmatrix}2 & 0 \ -9 & -1\end{bmatrix} = \begin{bmatrix}-1 & 0 \ 9/2 & 1/2\end{bmatrix}\]

Check the Invertibility of Matrix C

Matrix C is \(C=\begin{bmatrix}3 & 7 \ 3 & -1\end{bmatrix}\). Calculate its determinant:\[3 \cdot (-1) - 3 \cdot 7 = -3 - 21 = -24\]Since the determinant is -24, matrix C is invertible. To find the inverse, use the formula:\[C^{-1} = \frac{1}{-24}\begin{bmatrix}-1 & -7 \ -3 & 3\end{bmatrix} = \begin{bmatrix}1/24 & 7/24 \ 1/8 & -1/8\end{bmatrix}\]

Find the Inverse of Matrix D

Matrix D is \(D=\begin{bmatrix}7 & 6 \ 0 & 3\end{bmatrix}\). Calculate its determinant:\[7 \cdot 3 - 0 \cdot 6 = 21\]Since the determinant is 21, matrix D is invertible. The inverse is:\[D^{-1} = \frac{1}{21}\begin{bmatrix}3 & -6 \ 0 & 7\end{bmatrix} = \begin{bmatrix}1/7 & -2/7 \ 0 & 1/3\end{bmatrix}\]

Key concepts

Matrix Determinant

The determinant of a matrix is a special number that can give us insights into the properties of the matrix, especially when it comes to finding inverses. For a 2x2 matrix, such as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated by \( ad - bc \). This value helps us determine if the matrix is invertible. If the determinant equals zero, the matrix does not have an inverse. If it is not zero, the matrix can be inverted.

When you compute determinants, you are essentially measuring a "scale factor" for the linear transformation that the matrix represents. A non-zero determinant means that the transformation is reversible, which is why the determinant being non-zero is a key condition in finding a matrix inverse.

Matrix Algebra

Matrix algebra involves operations that can be performed on matrices. Two fundamental operations in matrix algebra are addition and multiplication, but inverses only come into play with multiplication. When discussing inverses, focus especially on 2x2 matrices, as they provide a simpler context to understand inverses.

To find the inverse of a 2x2 matrix, use this formula - \[ A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \] The process involves determining the reciprocal of the determinant and multiplying it by the matrix of cofactors to adjust the matrix into its inverse form. This adjustment facilitates reversing the transformation represented by the matrix.

Remember, order matters: the formula rearranges elements in a way that creates an identity matrix when multiplied by the original matrix.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It involves lines, planes, and subspaces but in multi-dimensional forms, extending into matrices. The study of matrix operations, determinants, and especially inverses, is rooted in linear algebra.

Understanding inverses is vital for solving linear systems. When we say a matrix is invertible in linear algebra, it means we can find another matrix that can return the identity matrix when multiplied with the original. The identity matrix \( I \) acts as the numeric equivalent of 1 in matrix multiplication. This principles allows us to solve systems of equations efficiently, which is crucial when dealing with multiple variables represented in matrix form.

Linear algebra provides powerful methods for computational tasks, useful not just in mathematics, but also in physics, engineering, computer science, and beyond.