Concept 1: Invertible Matrices

A square matrix is said to be invertible if there exists another square matrix of the same order such that their product results in the Identity matrix.

Definition

If \(A\) is a square matrix of order \(m\), and if there exists another square matrix \(B\) of the same order \(m\), such that:

$$AB = BA = I$$

then \(B\) is called the inverse matrix of \(A\) and it is denoted by \(A^{-1}\). In such a case, \(A\) is said to be invertible.

Key Properties and Conditions

1. Square Matrix Requirement: A rectangular matrix does not possess an inverse matrix because for products \(AB\) and \(BA\) to be defined and equal, \(A\) and \(B\) must be square matrices of the same order.
2. Non-Singularity: A matrix \(A\) is invertible if and only if it is non-singular, i.e., \(|A| \neq 0\).
3. Inverse of Inverse: \((A^{-1})^{-1} = A\).
4. Reversal Law: If \(A\) and \(B\) are invertible matrices of the same order, then \((AB)^{-1} = B^{-1} A^{-1}\).

Solved Examples (Invertible Matrices)

Example 1:

Show that \(B = \begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}\) is the inverse of \(A = \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}\).

Solution:

To prove \(B\) is the inverse of \(A\), we must show \(AB = I\).

$$AB = \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$$

Similarly, \(BA = I\). Thus, \(B = A^{-1}\).

Example 2:

Find the inverse of \(A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}\).

Solution:

Let \(B = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}\).

$$AB = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I \implies A^{-1} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$$

Example 3:

If \(A\) is an invertible matrix, show that \((A^T)^{-1} = (A^{-1})^T\).

Solution:

We know \(AA^{-1} = I\). Taking transpose:

$$(AA^{-1})^T = I^T \implies (A^{-1})^T A^T = I \implies (A^T)^{-1} = (A^{-1})^T$$

Example 4:

If \(A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}\), verify that \(A^2 – 4A + I = O\) and use this to find \(A^{-1}\).

Solution:

$$A^2 = \begin{bmatrix} 7 & 12 \\ 4 & 7 \end{bmatrix}, \quad A^2 – 4A + I = O, \quad A^{-1} = 4I – A = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$$

Example 5:

For two invertible matrices \(A\) and \(B\), if \((AB)^{-1} = B^{-1}A^{-1}\) holds, find \(B^{-1}\) if \(A=I\) and \(AB = \begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix}\).

Solution:

Since \(A=I\), \(B=AB=\begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix}\).

$$B^{-1} = \begin{bmatrix}1 & -2 \\ 0 & 1\end{bmatrix}$$

Concept 2: Uniqueness of Inverse

A significant theorem in matrix algebra states that if an inverse of a square matrix exists, it must be unique.

Theorem: Uniqueness of Inverse

Statement: Inverse of a square matrix, if it exists, is unique.

Formal Proof

Let \(A\) be a square matrix. Suppose \(B\) and \(C\) are two inverses of \(A\).

$$AB = BA = I, \quad AC = CA = I$$

By associativity and identity property:

$$B = BI = B(AC) = (BA)C = IC = C$$

Thus, the inverse is unique.

Solved Examples (Uniqueness and Properties)

Example 1:

If \(A\) is invertible and \(AB=AC\), prove \(B=C\).

Solution:

$$A^{-1}(AB)=A^{-1}(AC) \implies B=C$$

Example 2:

Show that the inverse of a symmetric matrix is also symmetric.

Solution:

$$(A^{-1})^T = (A^T)^{-1} = A^{-1}$$

Example 3:

If \(A^3 = I\), prove \(A\) is invertible.

Solution:

Let \(B=A^2\), then \(AB=BA=I \implies A^{-1}=A^2\).

Example 4:

Verify uniqueness using \(A=\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\).

Solution:

The proof logic holds for any square matrix.

Example 5:

If \(AB=I\), does it mean \(B=A^{-1}\)?

Solution:

Yes, for square matrices, \(AB=I\) implies \(BA=I\), so \(B\) is uniquely \(A^{-1}\).

Self Exercise (HOTS)

1. If \(A\) and \(B\) are invertible matrices of same order, prove \((AB)^{-1} = B^{-1} A^{-1}\).
2. If \(A^2 – A + I = O\), prove \(A\) is non-singular and find \(A^{-1}\) in terms of \(A\).
3. Let \(A\) be invertible. If \(A\) commutes with \(B\), prove \(A^{-1}\) commutes with \(B\).
4. Find inverse of \(A=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{bmatrix}\).
5. If \(A=\begin{bmatrix}2 & 3 \\ 5 & 7\end{bmatrix}\) and \(B=\begin{bmatrix}1 & 0 \\ -1 & 2\end{bmatrix}\), verify \((AB)^{-1}=B^{-1}A^{-1}\).
6. If \(A\) is idempotent (\(A^2=A\)) and invertible, prove \(A=I\).
7. For \(3\times3\) matrix \(A\), if \(A^2+2A+I=O\), find \(A^{-1}\) and show it is unique.
8. If \(A\) is skew-symmetric of order 3, is it invertible? Explain.
9. If \(A\) is invertible, prove \(\det(A^{-1})=1/\det(A)\).
10. Let \(A=\begin{bmatrix}1 & \tan x \\ -\tan x & 1\end{bmatrix}\). Show \(A^T A^{-1} = \begin{bmatrix}\cos 2x & -\sin 2x \\ \sin 2x & \cos 2x\end{bmatrix}\).