Inverse Matrices

Level 1 - Linear Algebra topic page in Matrix Algebra.

Principle

An inverse matrix reverses the effect of a square matrix under multiplication. A matrix with an inverse is called invertible or non-singular. A matrix without an inverse is called singular.

Only square matrices can have two-sided inverses.

Notation

\(A^{-1}\)

the inverse of A, when it exists

\(I_n\)

the n by n identity matrix

\(\det A\)

the determinant of A

\(ad-bc\)

the determinant of a 2 by 2 matrix \begin{pmatrix}a&b\c&d\end{pmatrix}

\(\mathbf x\)

an unknown vector in a matrix equation

The Core Test

An \(n\) by \(n\) matrix \(A\) is invertible when there is a matrix \(A^{-1}\) such that

Inverse definition

\[A^{-1}A=I_n,\qquad AA^{-1}=I_n\]

For a \(2\) by \(2\) matrix, use the determinant test:

Two by two determinant

\[\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\]

If \(ad-bc\ne0\), then

Two by two inverse

\[\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\]

If \(ad-bc=0\), this formula cannot be used because it would divide by zero, and the matrix is not invertible.

Worked Cases

Question

Find the inverse of \(A=\begin{pmatrix}2&1\\5&3\end{pmatrix}\).

Answer

First compute the determinant: \(ad-bc=2\cdot3-1\cdot5=6-5=1\). Since the determinant is non-zero, the inverse exists. Apply the formula: \(A^{-1}=\frac{1}{1}\begin{pmatrix}3&-1\\-5&2\end{pmatrix}=\begin{pmatrix}3&-1\\-5&2\end{pmatrix}\).

Examples

Question

If \(A\) is invertible and \(A\mathbf x=\mathbf b\), how do you solve for \(\mathbf x\)?

Answer

Left multiply both sides by \(A^{-1}\): \(A^{-1}A\mathbf x=A^{-1}\mathbf b\). Since \(A^{-1}A=I_n\), this becomes \(I_n\mathbf x=A^{-1}\mathbf b\). Since \(I_n\mathbf x=\mathbf x\), the solution is \(\mathbf x=A^{-1}\mathbf b\).

Transpose

Augmented Matrices